mathematical reasoning. Okay, mathematical reasoning. Let me tell you, this is one of the very easiest of all topics that you are going to get. Normally in competitive exams, especially if you talk about J main one question every year comes on this. Okay, and that is very, very easy. So I don't find anybody making mistakes in such topics. Okay. So, I'm planning to cover up this topic in today's session itself so it is not that big also. So I think one class is more than enough to finish off this topic. So let's get started. Let's get started. I think people have started joining in. So, mathematical reasoning is basically a vertical of maths which actually comes from logic. Okay, so logic was one of the ancient fields of mathematics, even before geometry. So there used to be some group of people calls who fist. Okay, they were actually, you know, disciples of Aristotle tales and all those people who used to roam around in villages. You know, trying to, you know, pass on certain statements logical statements which used to confuse people. Okay, which used to baffle people. And that's how they used to, you know, earn their source of whatever income they had. Okay. But later on this field of mathematics was standardized and nowadays it is used in the field of electronics and computer science. So some of you would have done logic gates in computer science. In fact, when you study electric electrical and electronics engineering later on, let's say some of you would like to offer it. You would be coming across logic gates. Okay, so all those fields of engineering and computer science, they are basically based on this concept of logic. So mathematical reasoning is one of the, you can say application is to prove certain statements by the act of deduction. Okay, so there are two ways normally we prove statements in mathematics. One is by induction, which I'm sure you would have done in school principle of mathematical induction that works on induction principle. So induction principle is you, you basically work on specifics to prove a generic thing. Mathematical reasoning you go the other way around you use the generic, you use the generic thing to prove a specific thing. Okay, that is called the art of proving a mathematical statement by deduction. Okay, so there's one induction is another deduction. However, this is just just a history part of it and the the in general know how of what is mathematical reasoning. So primarily in this topic, we are going to talk about statements. Okay, we are going to talk about statements and towards the last part of the topic, we are going to even validate statements. So first of all, I would like to know from you, what is the difference between sentences and statements. So what is the difference between a sentence and a statement. Can anybody reply that. Yes, this Boolean algebra is definitely is a part of it, you'll realize it later on. Boolean maybe you can say it's an advanced version of it. But yes, it has obviously been inspired from the concept of logic. Yes, what's the sentence. So sentence is nothing but it's a meaningful collection of words. Isn't it. It's a meaningful collection of words. Okay, so collection of words which has got certain meaning attached to it that is called sentence. Okay. Sentence could be of various types. Okay, I can tell has given ideas statement is a fact. Okay, we'll discuss that out. So they're falling types of sentences that we normally use in our day to day communication. Okay, so types of sentences. There's something called assertions or assertive sentences. Okay, assertive sentences, assertive sentences are nothing but assertions or propositions, where you're trying to assert something or where you're trying to propose something. Okay, for example, if I say three is an odd number. This is something which I'm asserting. Isn't it, or I'm proposing that three is a prime number. Okay, for example, I can say, let's say earth is a star. Earth is a star now I may be wrong in my proposition also for example if you see earth is a star is a wrong proposition, or it's a false proposition, but nevertheless it's a proposition. Okay, so any assertive statement is like a proposition where you are proposing or asserting something. It could be two false or it could be open ended also for example if I say x is greater than seven. Okay, so I'm proposing x is greater than seven. Now, here we don't know whether it is true or false depending upon x value we can take a call, right. So in these three examples which I've taken you are trying to assert or you're trying to propose something so this is called an assertive sentence. Another type of sentence that we normally use is imperative sentence. Okay, what's an imperative sentence or imperative sentences imperative sentences are those sentences where you make a request or a command. Okay, where you make a request or you command something. So normally when we talk, there are certain cases, there are certain instances when we request something. Right, for example, please give me a glass of water. Right, you're making a request. Okay, glass of, of water, David to so far in your query. That's what we say Karna, am I correct. David to so far near you. Okay, or you're giving some command to somebody, switch on the lights. Okay, switch on the lights. Okay, so this is a command that you're making. Another type of sentences that we use is interrogative sentence. Where you try to interrogate somebody or you're trying to question somebody. Okay, so this is a sentence where some kind of a question is involved. For example, how much did, how much did you score in your duties? How much did you score in your last duty? In your last duty, so your examples are also like, you know, UT and obviously UT is a very important event for most of you. By the way, this is, this is a mock, which I'm making because many of you miss classes because of UT's. Okay, something could be like, where were you last Sunday? Where were you last Sunday? Okay, so these are sentences where you're asking a question. Okay, of course it has a question mark. And another type of sentence is your exclamatory sentence. By the way, I'm not trying to teach you English over here because I'm no English teacher here, but these type of concepts are actually asked in the competitive exam. Okay, they will say, they will ask you to, you know, write down the type of sentence being used in a certain state, in a certain particular question. Okay, exclamatory sentences are those sentences where you are trying to express some kind of a wish. Okay, you're wishing somebody or maybe you are expressing some kind of a joy or sorrow or any kind of an excitement. Okay, for example, what a wonderful day. Now, here you're not asking you asking any question, you're not asking what a wonderful day. You're just saying you're very happy that it's a wonderful day. Okay, or alas, we lost our CDS. Okay, I hope some of you who are reading the news. I think the day before yesterday, we lost the chief of defense staff. Mr. Wipendawar, very sad instance. Okay, in a helicopter crash. So this is a sorrow, which I'm expressing. Okay. So these kind of these are the type of sentences that we normally use in our day to day communication. So what is a statement then. So this is sentence. Okay, four types of sentence are there normally. So what's the statement then. So somebody said statement is where you are stating a fact that is not correct actually. So what is the statement, let me write that down. Statement are those sentences. So statements are those sentences. Or let me write statement singular statement is a sentence, a sentence. Of course it is a part of a sentence so sentence is a bigger set statements are subsets of it a sentence, which can be answered with a true or a false, but not both. But not both and must be unambiguous and must be unambiguous. That means in any sentence if you can answer that sentence by saying a true or a false just one of them, not both. You can't say sometimes true sometimes false or it could be both said no. So those are not statements. So statements are those sentences where you have only one answer for its truth value whether true or false, but not both. And there should not be any ambiguity ambiguity means there should not be any uncertainty in the way that sentence has been written, which makes you difficult to difficult to answer it with a true or a false. So please understand here. A statement follows law of excluded middle. It can either be true or could be false. It cannot have both the status. And there must not be any ambiguity about it. Okay. Now a few things I would like to add on over here. If there is a statement where you feel that there is a variable involved. Okay. That means because of the variable you are not able to say whether it is true or false, then those those sentences I would say I would say not those statements those sentences are actually called as open statement. Open statement is a word which is basically used for those sentences, which could have become statement provided the unknown involved in that sentence is known to you. Right. For example, if I say there is life, or you can say he is a good person. Okay, he is a good person. Now, you don't know whether that person he who is he who is this he are you talking about Mahatma Gandhi. Right. Or are you talking about Daoud Ibrahim. Who is this he. So till you know who is this he you will not be able to say a true or a false. Correct. So in that case it becomes a open statement. Many people says that open statement means statement only now. No, open statement means something which is eligible to become a statement. It is open to become a statement open means ready when it could become a statement if you make that small change in that sentence by removing that variable. Okay. Sentences which are both two and false are actually called paradoxes. Yes, absolutely. Okay. This statement. This statement is false. Why it is a paradox. It's a statement. So what is a paradox paradox is where you have both the chances of it being true or false. So those are not statements those are called paradox. Okay, anyways, we are not going to talk about paradox etc. But very important thing to understand is what is an open statement. Open statement are those sentences where there is a variable involved or there is an uncertainty involved because of that uncertainty. We cannot say that statement is true or a false. Right. Another example is nice. Let's say if I say there is life beyond earth. Let's say I make the statement. Okay. There is no position to say true to this. Has it been found that there is life beyond earth. But if we have not found it, does it make us eligible to say that there will, there is no life at all since you have not found it. Okay, so we cannot say a true or a false word because there is uncertainty about it right now. So this is an open statement. The moment we realize that there is a planet or there is a celestial body where life is existing then that would become a statement. Okay, nice. Let's go back and analyze which of these four sentences are eligible to be called a statement eligible to be called a statement. Let us understand from the last one. Is your exclamatory sentence eligible to become a statement. Ask yourself. So you need to ask yourself, can I answer these questions with the true or a false, but not both. And are they unambiguous. So if you realize what a wonderful day you cannot say a true for it. Right. There is no true and a false answer to it. What a wonderful day is somebody's, you know, you can say he's expressing his happiness. Okay. Of realizing that the day is a wonderful day. There's nothing true or false about it. Right. So this is not a statement. This is not a state. This is not a statement. Okay. Now what about interrogative sentence. Where were you last Sunday. Can you answer this with the true. Can you say a false. No, right. You will have to tell where were you right you have to give an answer in terms of a place right I was at my home. Right. Or I was in the playground. Right. So there's no true and false to this. So this sentence is also not a statement. Let me write it in white. I didn't be consistently white. It's not a statement. Okay. There is some question coming up. So the below statement is false. The above statement is true. So pair of statements you're making. Okay. Anyways, we'll come back to that. Now, imperative sentences imperative sentences is where you are making a command or a request. So this also you cannot answer with the two or the false. Give me please give me a glass of water. Will you say true for it. I don't think so you'll say yes, I will give you or you'll say, No, I will not. Okay. So there's no true and false to this. So this is also not a statement. Okay. Now coming to the first one assertive sentence. Three is an odd number. You'll say yes true. Okay, so you can answer this with the true. Earth is a star. You'll say no false. Earth is a planet. Seven X is greater than seven. Now this is this is ambiguous because there's a variable involved. So you cannot answer this with the two and a false, you know, one of them will be there, depending upon X value, either it will be true or it will be false. Correct. But since you don't know X right now, you can choose between two and a false.  Okay, so this is an example of an open statement. Okay, you cannot say it's a statement. It's an open statement, by the way. Okay, so these two are eligible to be called as statements. So basically please understand those assertive sentences where there is no ambiguity. Those are the only ones which we can call as statements. I'll repeat once again. The assertive sentences where there is no ambiguity. Those will be only called as statements. It's called open statement. Okay, it's not an open ended sentence. It's an open statement. Okay, but it is not a statement. The word open statement is basically used because it is open to become a statement. See, the word open has come from the fact that he's open to come. We say in our normal talking, when we say, will he come to the party? He's open to come. If you call him, he will come. Right, he's open. Open means he will agree if you ask him, but he may not even come, but he's open. Okay, so basically this is not a statement, but it could become a statement if you provide some X value. Can you just find any questions? Any questions? Any concerns? Okay, statements are normally symbolized by a small alphabet. Normally we use P, Q, R, S, T, etc. These alphabets that we use, they are called logical variables. They are called logical variables. Okay, because they act like the variables in the field of logic. Okay, because we can have different types of statements being used, different statements being used in the same question. So in order to name those, we use variables. So they are called logical variables. To give an example, let's say 2 is a prime number. Okay, so this could be a statement. So this is an assertion and you will say you can always put a truth value to it. So in this case, it is a 2. Right, 2 is a prime number. We all know that. So this is the way we write a statement. Okay, if you would have done mathematical induction chapter, you would write P and a bracket N. Right, used to feed it an input N, because that N acted like the variable name involved in that particular statement. Now there's something called negation. Let me go to the next slide. Negation of a statement. Negation of a statement is exactly the opposite of what the statement says. Okay. For example, if I say 2 is a prime number. 2 is a prime number. Its negation will be represented by this symbol P preceded by a tilde symbol. The symbol you can find it on your keyboards also many a times. So this is called tilde symbol. So this will say opposite to what P says. That means it will say P 2 is not a prime number. There are various ways in which you can write it. You can say it is not the case that 2 is a prime number. Okay, this is also a fine. Okay, or 2 is a composite number. Okay, so there are many various ways in which you can express it. Okay, so negation has the opposite truth value as what P has. So this is a typical truth value. So this is a typical truth table. Many of you would have used truth table in your junior classes. In fact, in computer science, you use truth tables. Okay. Truth table is a table. Okay, let me just write it down for you all. What is the truth table? A table definition of a truth table. It's a table showing the truth value. Truth value means true or false. Truth value means true and false. Many people say, so truth value means only true. No, just because the name has truth in it. It doesn't mean truth value means only true. So a table showing the truth value of a statement or a statement pattern or a WFF. Well-formed formula. Okay, now normally we use WFF for a complicated statement. Okay, WFF. And it's component statements. And it's component statements. Component statements. That table is called a truth table. How many of you have seen truth table earlier on? I'm sure people who have taken computer science would be using. Yes, negation works like the not operator. Right. Okay, many of you have seen. Okay, this year you have seen. Okay. So let's write down the truth table for negation of or not of a statement. So P if it is true, its negation will be false. And if P is false, its negation will be true. Okay, so it basically acts like the not gate. Okay, so what is the truth table? It is clear. What is negation of a statement that is also clear. Now when it comes to negation, a very important type of question is asked to find negations of certain statements which start with a quantifier. Let us learn that first, and then we'll talk about how do we find out negations of statements which start with quantifier or which has some quantifier inside it. So when it comes to quantifiers, there are two types of quantifiers that we normally talk about. One is called existential quantifier, existential quantifier. And another is called a universal quantifier, existential quantifier, and another is called a universal quantifier. Okay. Now what is an existential quantifier existential quantifiers are basically phrases like, they are phrases like, there exists. There exists. I'm sure you would have seen a lot of statements containing this phrase, there exists, am I right? There exists a prime number which is even, isn't it? Which after you have seen there exists and all. Yeah. Or statements which begin with phrases like, at least, okay, at least, at least, at least dot, dot, dot. For example, if I say there exists at least one planet which has life on it. Okay, so that at least word is basically acting like an existential quantifier. The word has come from exist, existential has come from the word exist. So you are basically trying to show that there exists certain quantities which basically have that characteristic as mentioned in the statement. Okay. Or some. Okay. So even some. So some odd numbers are prime. So that word some is acting like an existential quantifier. So what are universal quantifiers universal quantifiers are phrases with sound like this for every or every whatever you want to call it. Okay, for every x belonging to real number x square is greater than equal to zero right so that's for every basically acts like a universal quantifier. Or all. Right. For example, if I say all equilateral triangles are isosceles. See, by the way, when I'm making a statement, I need not always make a true statement. My statement can be false also. Nevertheless, it is a statement. So many people say it is false know how it is a statement statement can be true statement can be false also. Okay, so don't always expect me to make true statements. Okay. So even words like none or no none, etc. These are all these are also universal quantifiers because they are, they are taking the entire universe into their, you know, per view. For example, a statement like this. No one in Germany speaks German or no one in Germany speaks English let us let's make some, you know, no one in Germany speaks English. Okay, so here I'm just trying to cover every person living in Germany under that statement. Right. Are you getting my point. So these are all called universal quantifiers. Okay. When it comes to negation, when it comes to negation of statements which begin with universal quantifier or existential quantifier. I've seen many people making mistakes. Okay, now let me see whether you are making the same mistake or not. Okay, so let me ask, let me ask a question to you. Let's see who are who are able to answer this. So I have a statement here. I have a statement here. All natural numbers are negative. All natural numbers are negative. Okay. Can anybody tell me what would be the negation of this statement. Write it down on the chat box. I would like to see your response on the chat box. All natural numbers are negative. Sorry, that's wrong. Satyam, not completely right, but you have made a good attempt. Aria, wrong. Harshita, wrong. See I told you know many people make mistakes. Many people make mistakes. Most of you are saying no natural number is negative or all natural numbers are not negative. Okay, now here please understand all natural numbers are not negative to a certain extent it is clear, but your sentence framing is not that correct. Okay. So here please note that. Yes, Aryan, correct. Here you know that your statement is starting with a universal quantifier. Okay, so this is a universal quantifier. Please note that whenever a statement starts with universal quantifier, even one instance of the population that you're referring to, if it is, you know, if it is not following what the statement is saying, then also the statement gets negated. For example, if I say all swans are white and tomorrow you come and show me said, see this swan, this is black in color. So what happens to the statement which I gave you, it becomes false, right? It says that your statement was false. I got a swan which is white. So I did not show every swan to you need not show me every swan existing in this earth to show me that they are black or to show that they are not white. So even one instance you get where you realize that whatever I have said is not being followed. Then that negates that statement. That is why I keep telling you every time that if you want to disprove something, one example is sufficient. But if you want to prove something, you need a generic proof. So here I'm trying to say all natural numbers are negative. That is my statement. If it is sufficient to say that there exists a natural number which is non negative is good enough to negate my statement. Okay, so please note, if there is a universal quantifier in a statement, its negation would contain existential quantifier. So you could answer this question. In the following ways, you can either say there exists a natural number, which is, which is non negative. You need not say positive because zero could also be, which is non negative. Or you could say at least, at least one natural number, one natural number is non negative. Okay, is non negative. Or you could also use some natural numbers, some natural numbers, numbers are non negative. If one of them will fetch you the marks, they're all the right answers. So they're all correct. Okay, but never say no natural number is negative. That is not going to work out. Your teacher will mark it wrong. Okay. Similarly, if I give you a statement which has got existential quantifier. For example, let's say another example I'm taking R is a statement which says there exists a prime number, prime number, that is even. That is even. What do you think would be the negation of the statement? What do you think is the negation of the statement? There exists a prime number that is even. Neil, there exists a prime number that is even and all prime numbers even have something in common. They're not complimentary. They're not negation of each other. There exists a prime number that is even. No, we are not understood. There exists. There exists has an existential quantifier. There exists a prime number that is even. You need to tell me the complimentary of that. Either you say there is no prime number that is even, or you say all prime numbers are odd. Okay, both are fine. So please note, this will start with the universal quantifier. So if there is an existential quantifier in the statement, its negation will have a universal quantifier. Either you say all prime numbers are odd, or there is no prime number which is even. Are you getting the point? Are you getting my point? So please understand this fact. I like one more answer to this. Either of the two you can use. There is no prime number. So this no acts like a universal quantifier. There's no prime number that is even. Okay. So you are making a universal statement here. Okay. So please understand, please understand here. The moral of the story is, so please note this down. The moral of the story is, if there is a statement which is having a universal quantifier. Okay. It's negation. It's negation will contain a existential quantifier. Okay. And vice versa. That means if a statement has a existential quantifier, it's negation will contain a universal quantifier. It's negation will contain a universal quantifier. Is it fine? Any questions? Any questions? Any concerns here? There was a question asked, I think in one of the exams. Okay. We will take that question, but before that I would also take a few basic ones. Let's, yeah, let's start with this question. Okay. First of all, let me start with classification of classification of sentences. So please categorize the following, the following sentences. Please categorize the following sentences. So please use the following every variation. Don't write everything. If you think a statement is a sentence is an assertive sentence. You write an A for it. So please use the word alphabet A for assertive. Please use I am for imperative. That means a command or a request. Please use IN for interrogative. Any type of questioning is happening. And please use E for exclamatory. So any kind of a wish or any kind of an expression of sorrow, excitement, joy, et cetera. So first one, please put the question number. First one. What do you think? A, I am, I am or E, which category does it fall in? Which category does it fall in? A, very good. So this is an assertive sentence because you're asserting. Chandigarh is a state. Okay. I mean, it's a false assertion, but nevertheless it's an assertion. Okay. New Delhi is the capital of India. Assertive. 23 is a prime number. Assertive. Rational numbers are real numbers. Very good. Assertive. There's always a real number. Sorry. There's always a real root of any quadratic equation. Assertive. It may be wrong also because, you know, you need not have all the real roots to a quantity. Okay. Assertive. Very good. 15 S C pair numbers. Assertive. 40 is the absolute eighth one. Anu bring a glass of water for me. It's an imperative sentence where you are giving a, you know, command to anu. Where are you going? That is interrogative. Correct. Bravo. You fought very well. It's an exclamatory sentence. Okay. Is it fine? Anyways, these are just the basic school level type of questions. Let's take. Yeah. let's take this one write down the negation of the following statements let's start with the first one chinnai is the capital of Tamil Nadu what is going to be the answer right plain and simple don't you know complicated by using unnecessary terms so here you can say chinnai is not the capital of Tamil Nadu chinnai is not the capital not the capital of Tamil Nadu okay second one root two is not a complex number so what will be the negation of it root two is a complex number very good okay next one all triangles are not equilateral now here be careful why be careful because your statement contains a universal quantifier okay so all triangles are not equilateral okay yes you're claiming that no there's no equilateral triangle existing yes so you can answer this in various ways you can say there exists there exists a triangle which is equilateral okay there exists a triangle which is equilateral there's several ways to answer it okay you can also answer it like this at least one triangle is equilateral at least one triangle is equilateral okay or you can answer it like this some triangles are equilateral some triangles are equilateral again Arya you're making the same mistake all triangles are not equilateral its negation will start with existential quantifier it should have an existential quantifier what is this which I wrote what is this which I wrote statements having universal quantifier negation should have existential quantifier okay so all triangles are not equilateral the negation is not all triangles are equilateral or every triangle is equilateral no even if one triangle becomes equilateral it is good enough as a negation for it are you getting my point what I'm trying to say next the number two is greater than seven what is the negation yeah the number two is not greater than seven good enough this is a simple statement nothing to worry about it doesn't have a quantifier involved but yes fifth one again you have to be careful there is a quantifier sitting over here every natural number is an integer correct sattam there exists a natural number which is not an integer there exists a natural number that is or which is which is not an integer excellent okay or some natural numbers are not integers or at least one natural number is not an integer all of them are perfectly okay is this fine any questions any questions now let me give you the question which actually came in one of the competitive exams the question was like that this okay let's say s s is a non-empty non-empty subset of real numbers okay and there is a statement which says there is a statement p which says there is a rational number there is a rational number x belonging to s such that such that x is greater than zero okay which of the following is negation of p which of the following which of the following is negation of p options there is a rational number there is a rational number x belonging to s such that such that x is less than equal to zero option b there is no rational number x belonging to s such that x is less than equal to zero every rational number x belonging to s is satisfying or satisfies x less than equal to zero and finally x belongs to s and x is less than equal to zero implies x is not rational okay which of the following do you think is the answer to this question I'm putting the poll on if you want you can respond on the chat box as well but I would also like to see your response on the poll so that I know what is the count of the people answering this question okay I have got eight responses so far sorry 10 responses so far okay should we stop the poll in the next 15 20 seconds five four three two one so most of you have said option number c okay d is of course not the answer it is something else or d is rule out now let's try to see could a be the answer see this is there is already a existential quantifier there is means existential quantifier okay the word exist need not always exist okay if you read this you'll automatically know that there is something is there that means there exists something okay so this is an exist existential quantifier so your negation of it must contain a universal quantifier like no or every like that okay so first one there exists a real number which is less than equal to 0 sorry rational number which is less than equal to 0 this cannot be your answer so those who are answering with a sorry this cannot be your answer b there is no rational number such that x is less than equal to 0 basically it is saying trying to say the same thing that there exists a rational number greater than 0 right are you getting one point so option number b is basically having it's basically talking about certain numbers which are greater than 0 which is almost rhyming with what p is saying isn't it so when you say there is no rational number lesser than equal to 0 that means there are some rational numbers which are greater than 0 okay in fact you're trying to say that all of them are greater than 0 so there is and all of them are greater than 0 okay which basically rhymes with each other so this cannot be the complementary right complementary has to be exactly Ulta it cannot have some overlap with it like the way you have in sets complementary set and the set cannot have an overlap are you getting my point so this b option and this p statement will have some overlap so this cannot be a this cannot be a complement of or negation of that statement but if you say every rational number is less than 0 then this is exactly opposite of what is this okay so if you're saying every rational number is lesser than equal to 0 it cannot have any overlap with there is a rational number greater than 0 so option c is the right option are you getting my point this problem came in one of the j main exams I think in 2010 or 11 it came is it fine so please do not take the concept of negation slightly some very good questions have been framed on this now with this we are now going to move towards the concept of compound statements compound statements many of our concepts will be coming from this topic itself of compound statements so what is a compound statement so basically a statement let me define it first a statement made of a statement made of simpler statements so if more than one simple statement has been connected in some format what are those connectors we will discuss it in some time so the statement which is made up of some simpler statements those are called compound statements these simpler statements are actually called the sub statements also called as the component statements many a times also called as component statements okay so what are the compound statements that we are going to talk about primarily we are going to discuss only three types of compound statements so what are the types of compound statements that we are going to talk about let's talk about them we are going to talk about first statements which are connected by and operator which is called conjunctions okay so these are statements which are connected by the and operator okay we'll talk about it in detail secondly we are going to talk about disjunctions also called alternations these are those compound statements which are connected or these are those compound statements where your sub statements or component statements are connected by or operator we will talk about this also and the last type of compound statements that we are going to talk about is called are called implications so there are two types of implications that we are going to talk about if then implication and if and only if implication okay by the way if then implication is also called conditional conditional statements okay remember in your coding you have some conditional blocks if then block that's actually called a conditional block yes or no and if and only if is called by conditional statements by conditional statements okay so primarily these are the three types of statements that we are going to you know cover up conjunctions disjunctions implications of all the three this is the most important one okay this is the heavier of all the three so please note this down and we will start with conjunctions now do let me know once you're done with copying then everybody done with the copying okay shall we go to the next slide wait i know i know done okay then aria okay conjunctions so conjunctions are those statements where they are two simpler statements or they could be more than two also i'm just giving a simplistic example they're connected by and operator so and is basically a logical connector okay we also call it as a as a operator also logical connector slash operator symbolically we represent this entire thing by this symbol p q okay now most of you would be thinking said this is very simple similar to that intersection a symbol which we use in our sets yes in our sets if you used to have us i can say you know downward opening you can say no structure but it's a slightly sharper one now from where these symbols have come actually they have been borrowed from ancient times okay normally if you see if you read very ancient greek literature and all they used to use some symbols to represent something right so for male female child they used to have some different different types of symbols okay so that has been taken away from there i would not go to the historical part of it so some examples which i can give here is let's say if i say 10 is divisible by 10 is divisible by 2 and 3 okay 10 is divisible by 2 and 3 so if you see this is a compound statement okay and is made up of two simpler statements what are the two simpler statements here 10 is divisible by 2 10 is divisible by 2 and 10 is divisible by 3 10 is divisible by 3 okay so you have connected them by a and connector but mind you not all the statements which contain and becomes a conjunction okay at many places and doesn't like act like a connector can you give me a statement where and doesn't act like a connector can anybody give me a statement where and doesn't act like a connector anybody need not be mathematical okay let me give you fire and water do not go together do not go together okay so here and if you remove let's say if you read it like this fire do not go together water do not go together it doesn't make any sense right so those statements would have would not qualify to become sentences also because they don't don't can wear meaning okay so here and is a must it is not made up of two component statements okay another example i can give you is let's say ram and sham our friends ram and sham our friends so here this and is not acting like a conjunction okay this is not a conjunction so let me write it down the following are not conjunctions the following are not conjunctions another example which i can give you is let's say i watch this movie i watch the movie movie mr and mrs let me write and here to be more specific mr and mrs kanna okay so here if you see the name of the movie itself contains and it is not acting like a connector you can't say i was the movie mr kanna and i was the movie mrs kanna mr and mrs kanna is the name of that particular movie it's the title of that movie so so these are not conjunctions so please do not start claiming every statement which has got and into it as conjunctions okay now what are the truth table of this conjunction many of you are already aware of it but i'll still make it for you so let's say p q p and q the truth table goes like this by the way when you have two component statements how many entries or how many permutation combinations of truth values you require over a four correct so you can have true true true false false true false false so four inputs are required okay so how does this four input come up very simple every component statement can have two truth values every component statement can have two truth values either it would be true or false so if there are two component statements p q then you'll need four combinations of these truth values two true true false false true false false if there were three component statements then how many combinations would have been needed eight if four component statements would have been there then how many combinations would have been 16 okay so the formula is two to the power n okay anyways so this is the truth table this is how a compound statement works if both the statements component statements are true then the compound statement p and q is true if one of them is false it is going to become false okay so only when both of them are true so the only occasion when a conjunction will be true is when both the component statements or the sub statements making that conjunction are true okay this is already known to you so I will not be wasting that much time even the people who have taken bio I'm sure you would have done this in your junior competitions class 9 10th haven't you 1 into 1 is 1 0 into 1 is 0 0 1 into 0 is 0 0 0 0 that same concept is here is this fine any questions any questions any concerns related to our conjunction is this fine can I go to the next one disjunction so the second type of statement compound statement that you normally get to see in your competitive exams are disjunctions also called as alternations okay so what is a disjunction disjunction is when the two component statements are connected by or connected okay but now please note that there are two types of or connectors so please note or connectors are of two types one is called exclusive or okay and another type of or operator is inclusive or operator now what is an exclusive or exclusive or basically means if one happens the other cannot happen okay so they can happen only one at a time for example if I use a statement having exclusive or it would sound something like this a number is either positive or negative as you can see this is an exclusive or because the number cannot be positive negative at the same time yes or no yes or no either the coffee either the coffee is hot or cold cannot be hot cold simultaneously okay or you can also say like this the students are in class or in playground okay so as you can see here the students cannot be simultaneously in the class in the playground so this or is acting like a exclusive or so if one happens the other will not happen are you getting my point clear whereas inclusive or is where both the options are possible for example let's say either the bus is not moving or my watch is slow okay so let's say you're traveling somewhere and you realize that you know you have not reached the destination so you will say either my is the bus is not moving or my watch is there's some problem with my watch okay so both can happen actually simultaneously the bus may not be moving and your watch may be slow also right so one doesn't exclude the occurrence of the other are you getting my point so both can also happen right so union you can say yes union is basically a type of an inclusive or because every element either can have fall in A or fall in B or in both of them right Satya okay another example I can give is a I would prefer I would prefer tea or coffee okay that means even if you bring both of them to me I can have both of them okay it's not like I feel drink tea I cannot drink coffee I can be if I drink coffee I cannot drink tea I can have both the both the you can say beverages is it fine now our discussion will be around this we will be not discussing this this is not in our syllabus okay we will only restrict our understanding to inclusive or so inclusive or is symbolized by is symbolized by up arrow sorry down arrow okay so the down arrow is used to symbolize or a connector between two simple statements up arrow is used to signify and connector between two simple statements is it fine up arrow down arrow please be clear about it so there is a way to remember it also union if you just try to make the you know you sharp like this okay or intersection if you'd like to like like this okay so this acts like a inclusive or and this acts like your and okay so this is how normally I also used to remember when I was in my preparation days right so what are the truth table of an inclusive or for an inclusive or the truth table goes like this let's say these are the four status okay I found this on the web for these are the four so true or true is true true or false is true false or or true is true false or false is false that means the only occasion when our disjunction becomes false is when both the component statements are false is this fine any questions any concerns okay many examples can be cited I would just like to cite one example for a conjunction sorry disjunction example let's say I say 15 is or 10 is divisible by 2 or 5 okay that means it could be divisible by both also okay so inclusive this or is an inclusive or even though we do not explicitly write it is it fine okay now with this I will be running you through certain algebra of sets and certain terms which are going to be very important for us so let us move into that and then I will move towards the idea of implications okay so I'll be now taking you yes sure I will show the truth table so now I will be taking you to certain algebra of statements or rules of statements okay and then I will be moving towards the concept of implication then everybody so algebra of statements now when I am giving you this algebra of statements you would all be reminded of your set chapter okay in set chapter more or less the same algebra is applicable so you will hardly find any difference okay that's why it becomes very easy for us to remember them also so the first one is idempotent law idempotent law almost like the same idempotent laws that we learned in our sets so if you take a statement p and and it with another statement p that means you take a conjunction of p with itself it is going to give you p itself now all of you please make note over here I have used a special symbol over here okay this symbol let me just talk about it before I proceed this symbol or many times books will use this symbol or in some cases they will use this symbol also okay but normally this symbol is a symbol which is going to be used later on so let us not use it right now so these two symbols are normally used to signify logical equivalence the meaning of logical equivalence means having the same tooth table so in the eyes of mathematics two statements are logically equivalent if they have the same tooth table that means I claim that if you make a tooth table for this and you make a tooth table for this they will exactly be the same that is why these two statements are logically equivalent or these two wffs the well formed formulas they are logically equivalent to each other so you will see the symbol being used multiple times by me okay if they are not logically equivalent normally we cross it after writing it we cross it that is called not logically equivalent okay for example if I write something like this three dashes many times it is used for identity also if you open your international books you'll see that they use this for identity so they will say cos square theta plus sine square theta they'll put three dashes and put a one that means whether you say cos square theta plus sine square theta or whether you say one they are both same they mean the same thing that is why they say it's their identity also so if you cross it it means they are not logically equivalent not logically equivalent okay is this fine so this is something which I'll be using time in again I hope there is no confusion about that okay so p and p is logically equivalent to p please do not write equal to anon in a case of statements we don't write equal to equal to doesn't make sense equal to is used for numbers okay numbers or maybe the you know expressions in your algebra in logical statements we don't use equal to similarly p or p is logically equivalent to p okay so these two laws are called the idempotent laws note this down the second law is associative laws so what are these associative laws just like you have learned in your sets chapter if you do p and q okay and r it is logically equivalent to this okay that means it doesn't matter whether you associate p and q first and then take an operator with r right so even if you do q and r and first and then take anand with p the result is still the same in fact the order of anding doesn't matter okay in any order you random order also you can pair them up similarly goes with disjunction also p or q or r is logically equivalent to saying this okay see all of these you have already done it is just a formal you know application of it two statements as well next is your distributive law distributive laws distributive laws also very much similar to what we had learned so you have a distribution of r over and so you can always write this like this okay or you could have a distribution of and over or same as what we had for union and intersections next next is your involution laws involution law so involution law is simple negation of a negation gives you the same statement back obviously isn't it so the negation of a negation will begin the same so if you do even number of negations you are back to the same statement yes or no so this is called involution law and de morgan's law which is something which you have already done in your sets that is yet again coming up in this chapter also de morgan's laws so de morgan's law says that if you take a if you take a negation of a conjunction it is logically equivalent to saying negation p or negation q okay just like just like we had a intersection b complement equal to a complement union b complement okay in the same way p and q negation is p negation q negation connected by or operator getting my point so and here becomes or applied between the negations getting the point similarly negation of p or q is logically equivalent to negation p and negation q just like we had a union b complement equal to a complement intersection b complement okay so this will help you to remember the results here is it fine any questions now what are the proof for this all of them can very be easily proved by tooth table okay can be very easily proved easily proved by use of tooth tables okay so you can make tooth tables for the left hand side and the right hand side and you will realize that they become logically equivalent is this right so these are the laws now I'll be telling you certain terms which will be coming across so first note this down and if you have any questions any concerns do let me know let us do one thing let us prove the fifth one just as a practice okay so let us prove De Morgan's law applied to statements by using tooth tables have you all copied this can I go to the next slide done copied okay so let's prove one of them just the first part a sorry p and q negation is logically equivalent to negation p or negation q okay so what I will do is I'll make a tooth table in this to table I will make p q negation p negation q p and q negation of p and q and finally negation of p or negation q so let me write it down so p q negation p negation q p and q negation of p and q and finally negation p or negation q that is your right hand side okay so there are two there are two sub statements here or two component statements here so I will require four inputs true true true false false true false false correct so if this is your p what is your negation p can somebody tell me write it down on the chat box what should I write false false true true okay what should I write for here false true false true very good what should I write for p and q true false false false what is the negation of it false true true true now take the disjunction of negation p and negation q or take the or operator between them so f or f is f f or t is t t or f is again t t or t is t what do you see here is that they have the same table no difference so basically it signifies that these two will have the same tt same tt means they are logically equivalent that means if you say negation of p and q it is as good as saying negation p or negation q got the point so basis of this let's take few questions let's take few questions is it clear the proof is clear by using toothbrush tables any questions here all right so let's take a small question based on the second the negation of 2 plus 3 equal to 5 and 8 less than 10 is which of the following so i'm putting the poll on i would request everybody to answer very easy question i'm giving you around 45 seconds okay satyam we'll see waiting for at least 70 percent of the people to answer come on guys this is not a rocket science you should all get it okay five four three two one go all right so 11 of you have responded only most of you have said option number c that's absolutely right so you can call this statement to be let's say p p is saying 2 plus 3 is 5 and you can call this statement to be q 8 is less than 10 so what you have to do is you have to write down the negation of p and q right because these two statements are connected by and operator so as per our de morgan's law it is negation p or negation q so how would you read negation p negation p means 2 plus 3 is not equal to 5 this symbol is standing for or negation q means 8 is not less than 10 so option number c is absolutely right any questions okay let's try one more these are all simple ones again 30 seconds to answer this question as well what are the negation of this statement ram is in class 10 or rashmi is in class 12 ram is in class 10 or rashmi is in class 12 yes almost one and a half minutes gone 5 4 3 2 1 go okay let me end the poll most of you have gone with option number b b for bangalore see these two statements are connected by or operator so the answer to this question is ram is not in class 10 and rashmi is not in class 12 correct so if i say p is a statement where ram is in class class 10 and let's say q is a statement where you're saying rashmi is in class 12 okay what you have to find out you have to find out negation of this correct and as per our de morgan's law it is negation of p and negation of q which means to say ram is not in class 10 and rashmi is not in class 12 okay i'm sorry ram is not in class 10 rashmi is not in class 12 okay now out of these four options which option do you think is the most appropriate ram is not in class 10 but ram is in class 12 ram is not in class 10 but rashmi is not in class 12 what is this but doing it should be and right so i don't think so you should mark this also now i mean if they would have said neither ram is in class 10 nor rashmi is in class 12 that also is acceptable that is also is the correct way of saying ram is not in class 10 and rashmi is not in class 12 okay but if you see c1 either ram is not in class 10 or ram is not in class 12 that is doesn't make sense so as per me the appropriate answer here should be none of these none of the above are you getting my point but you know please please apply your English bit of English knowledge also but not to a very large extent many a times the question setter will be very clear about the options it will not be very dubious if not be very we can say you know vague option given to you they are very clear about it okay so if you see your previous here questions on this topic their options are pretty pretty you know clear unambiguous okay all right now apart from this there are terms which you need to know so few important terms with respect to statements with respect to statements one term that you will keep on coming across is tautology anybody knows what's a tautology anybody has heard of this term tautology right sattva is correct so it's a statement which is always true statement which is always true so for a tautology if you see the truth table all the entries in the truth table of a tautology will always be 22222 okay for example something like this p or not p if you make a truth table for this let me make one quickly it will not take much of a time so let's say p not p and p or not p so let's say this is true false then this will be false to not true or false is true false or true is also true so as you can see this is always true this is always true and in that case this statement becomes a tautology okay second kind of term that you will commonly see is a contradiction contradiction many places you will see fallacy fallacy okay so contradiction or fallacy is a statement which is always false a statement which is always false which is always false a simple example of this could be p and not p however we'll be solving many questions where you will come across you know complicated examples also of tautology and fallacy okay so if I make a truth table again just like I made for the above one p not p p and not p so if this is true false this will be false to remember true and false is a false false and true is again a false so this is always false so this statement is a contradiction so basically you're assigned to say p and not p also something like saying 3 is the prime number and 3 is not a prime number so this statement will always be false because both are contradicting each other isn't it okay so p and not p is a contradiction or is a fallacy the third type of term that you will you know see is a contingency contingency is basically a statement which is neither a tautology nor a fallacy nor a contradiction a statement which is which is neither a tautology neither a tautology nor a fallacy that means sometimes it will be true sometimes it will be false it cannot be true true true always it cannot be false false false always it will have a mix and match of true and false something like p or q that's a contingency p and q that's also contingency some occasions it was true some occasions it was false so just remember like this if it is not a tautology or a fallacy it is a contingency okay the fourth term that you will get to here is a dual statement or a duality dual statement also called as a duality now what is the duality of a statement or what is the dual statement of a statement so let's say let me symbolize it let's say there is a statement which has got p and q as the component statements and of course this may have a lot of you can say connectors like and connector or connector etc then the duality of this is represented by s star pq okay this is a symbol that we use and how is the duality obtained duality is obtained by replacing by replacing and with or and or with and in your statement given to you okay so for example if I say let me give an example if I say can you write the can you write the duality of this statement okay so let's say if I say write the dual statement of the following okay so somebody asks you what is the dual statement of this so you will say p now there is a and connector so put a or there okay and there is a or connector so put an and there so this will become a this will become your answer for the duality clear very simple okay nothing to worry about it so some questions have been asked to write down the dual statement of a particular statement so all you need to do is change and with or or with and that's it don't have to change negation and all please note that many people start removing negations also okay one of the very interesting properties of duality is this property if you have a statement with component statements as let's say p q r etc I'm just writing three of them if you're taking the negation of it then please note that the negation will be logically equivalent to the duality with the inputs as negation of these component statements this is a very very important property please note this down some questions have been framed around this property in the comparative exams also so any statements negation is logically equivalent to its dual statement but written in terms of negation of the component statements one of the examples you have already seen while I was giving you the de morgan's law I'll show that again first copy this down negation of a statement is nothing but or it is logically equivalent to its dual statement but written in terms of negation of the component statements okay so now let me show this example to you do you remember when I told you de morgan's law what was de morgan's law de morgan's law gave you this correct correct let's say p and r you're taking a negation of it it becomes negation p or negation r okay now let's try to understand it like this if I gave you this statement just this statement what is the duality of this what is the duality of this guy p or q or r correct now now in this duality if you replace your this is your s let's say then this is your s star correct in this s star if you replace p with negation p and r with negation r sorry I wrote a q by mistake I should write an r normally I keep writing p and q only so yeah so when you do this what do you end up getting you end up getting negation p or negation r correct so what does it mean it means that this particular property is coming from this itself so negation of p and r is as good as negation is as good as the duality constructed with negation p negation r so if this duality I constructed with I replace this with negation p and I replace this with negation r I end up getting the negation of this guy are you getting what I'm trying to say okay let me give you another example let me give you another example so let us say let us say I give you a statement p or q and r okay and I ask you what do you think is the negation of this statement so what I will do is I will save a lot of my time first I will write the duality of this guy what is the duality of this guy what is the duality of this guy p and q or r correct and what I will do is I will replace all my p q and r with negations so it will become negation p and negation q or negation r got the point so this law helps you to save a bit of time when you are writing the negations of our wff which has got these and and or connectors in between okay if there are many of them you can immediately save your time on writing them is it fine any questions any concerns clear everybody it is clear all right let's take problems now that we have done enough amount of concepts we are good to start with problems okay let's take this question let p and q be two statements negation p or negation p or q and negation p and negation q is a option a tautology contradiction neither totally nor contradiction means a contingency or both tautology and contradiction of course d is a you know wrong option but let me just launch the poll and take your response the best way is to make a truth table so make a truth table for this statement or this wff and see whether the truth table is showing 22222 or false false false false or is a mix and match and then give your response okay two minutes gone I can see almost two and a half minutes gone I can see only five of you have responded I can give you 30 more seconds okay five four three one and the full out of eight of you who voted five of you say c and three of you say it's a contradiction okay so let us make a truth table so p so I can only see two component statements here p and q so p q negation p negation q negation p or q negation p and negation q and finally the original statement so p q negation p negation q then negation p or q and we have negation p and negation q and finally our statement so let's put some truth values true true false false true false false okay negation of it will be false false true true negation of this will be false true false true now you have to take this these two and take their take their negation p and q you have to take their or operator okay so their or operation will give you true false true now just erase this off now you need to take and between negation p and negation q so you have to add these two guys so false and false is false false and true is also false true and false is false true and true is a true now these two fellows you have to take and so true false false false false and you realize the last one becomes a true so because of this it will neither be a tautology nor a contradiction so it is actually c option it's a contingency it's a contingency okay please note this down is this fine any questions any questions any concerns all fine let's take another one let's take this question it came in v i triple e v i t triple e relaunch what is the negation of this statement what is the negation of this statement all right two minutes gone eight people have responded so i'm switching off the poll in the count of five five four three two one go all right so nine of you have voted out of his seven of you say option number b now for such kind of question you should always take the help of duality okay duality gives you the negation in a much faster way remember the formula which i gave you duality of any statement like this okay whatever is i mean i'm just writing two of them because there are only two component statements here is equal to the duality written in terms of negation of the component statement okay so let's use this so let's write down the duality of this so duality of this guy so s star first let us write that down this right let me write yeah s star of this will be negation p or q p or negation q correct so this is a duality this is the dual of this now in the dual you replace p with negation p so it will become negation negation p which is p negation q negation p again negation negation q will become q so this is your answer to this question which option does it match with which option does it match with option b exactly so b is the right option how much time it takes okay so just write down the dual of that given statement in the dual of that given statement how do you write the dual replace and with or with and and in the dual you replace your component statement with the negations so p will become negation p q will become negation q yes or no is it fine any questions let's move on to the next one let's take this one which is the dual of the statement since we discuss about dual also let's take this one also which is the dual of the statement i'm launching the poll it should not take you more than 30 seconds okay let me stop the poll now five four three two one go okay most of you have set option numbers b b for bangalore it's very obvious so p up arrow will be replaced with down arrow again up arrow will be replaced with down arrow nothing else will change again down arrow will be replaced with up arrow that means and this is going to be your answer which option matches with this option number b or matches with this so this is the dual of this statement are you getting my point okay all right so with this we are now going towards the third type of compound statement which we discussed which was called the implications and as i already told you there are two types of implication which is called if then implication so i'll be starting with my if then implication which is also called conditional statements okay now many times you would have you know use if then in mathematics if you drop a perpendicular from the center on to the chord then it will bisect the chord correct if a number is divisible by four it must be divisible then it should be divisible by two right so these kind of statements which you keep using keeps you know reading they are called conditional statements so something like this says if this then this by the way this is not only the only way to write a conditional statement many a time the same thing is written like this p implies q they're same things no difference whether you say if p then q or whether you say p implies q it's the same thing no difference and you'll be surprised to know this is also right as p is the sufficient condition for q p is the sufficient condition for q right so all these three means the same thing even one more way of writing it i will show you some books will also write it like this q is the necessary condition for p necessary condition for p okay they mean the same they all mean the same same meaning so let me give you some examples let me give you some examples then you will understand the relevance of using these different different phrases for signifying the same thing so if p then q or if somebody says p implies q or if somebody says p is the sufficient condition for q or if somebody says q is the necessary condition for p they all have the same meaning in mathematics put it down everybody so let me give an example if i give you an example if it is a sunday if it is a sunday then school is closed so please note that being a sunday is a sufficient condition for the school to be closed in the sense that if somebody comes and tells you hey today is sunday then what image will you get regarding the functioning of the school you'll automatically imagine closed gates school is closed correct so just mentioning a sunday is sufficient enough for you to make it out that the school is closed right of course provided this statement is true so i'm assuming under the assumption that whatever statement i've given it's a true statement okay now please note that q is not the sufficient condition for p that means if the school is closed it may not be a sunday are you getting my point so if somebody comes and tells you hey let's say i'm talking to aria he says aria your school is closed so does it mean aria that it will be a sunday no it could be closed yeah it could be closed because of so many reasons maybe you know new a new variant of corona is there in the market or in i mean in the town or in the city or it could be like yeah it could be like gandhi jnd holiday or it could be like there is a leopard in the town i think once one school was closed because there was a leopard inside the school okay so if i say it is a sunday it automatically it is sufficient enough for you to guess the school will be closed but if i say the school is closed it may not be a sunday are you getting my point but now the school being closed is a necessary condition for it to be a sunday that means if the school is not closed then it is not sunday are you getting my point here please understand this this is deep slightly deep to understand if the school is not closed if i come and tell you hey the school is not closed then today can definitely be not a sunday am i right because if it was a sunday the school would have been closed are you getting what i'm trying to say see i'll give you another example uh let's say if you have fever correct if you have fever you are definitely not healthy are you getting the point but if you don't have a fever that still doesn't mean you're healthy you could have some other problem maybe you know some other issues correct but having a fever in the body is a significant of the fact that you are definitely not healthy correct so fever becomes fever becomes a sufficient condition to say that you are not healthy are you getting my point but if you don't have a fever it doesn't mean you are actually healthy right so in the same way if the school is not closed it is a necessary condition that today will be not a sunday because if it were a sunday the school would have been closed but remember when i was talking about circles the other day what did i tell you in a circle the necessary condition is the coefficient of x square and y square should be equal right so if the coefficients are not equal x square and y square it is definitely not a circle but that doesn't mean if the coefficients are equal it will be a circle no it can be parable also it can be lips also we'll we'll see in the other corner section chapter so having the coefficient of x square and y square equal to one is a necessary condition for it to be a circle if that is not met necessary means very important if that is not met it will not be a circle for example another example i'll give you oxygen is a necessary element for my survival right oxygen is necessary for my survival that means if oxygen is not there i will definitely not survive but that doesn't mean if oxygen is there i will survive always no i can die because of other factors also maybe because of thirst food because of food right so oxygen is necessary necessary thing means if that is not there then the other guy will not happen right it is not sufficient that means if you give me oxygen i will always survive no i may die because of other reasons sorry maybe looking like a negative example right so if the school is not closed then it is not a sunday that is the meaning of the school is closed becomes a necessary condition for it to be a sunday are you getting a point are you are you clear about the difference between the words sufficient and necessary so necessary has to always be read in negation for example if i say it is a sunday then the school is closed then sunday is a sufficient condition because if it is a sunday it is automatically implying that the school is closed but if i say school is closed is the necessary condition for it to be a sunday means if the school is not closed then it will not be a sunday idea is clear idea is clear initially many people take time to understand this okay now symbolically we you know write this as p and we put a single facing arrow towards q so this is a implies single facing arrow okay so p implies q or p is the sufficient condition for q or q is the necessary condition for p they all mean the same thing no difference okay now there are certain words which you need to understand let's say if i call this to be r then there are certain terms which you will keep hearing time and again for example converse so far i'm sure you would have heard of this converse in class 10th do you remember your teacher used to always prove converse of a statement right i used to be you know very very agitated why the teacher is doing converse isn't they mean the same thing okay they actually don't mean the same thing we'll talk about it through truth tables so when i say there is a statement if p then q then the converse of it will be if q then p now tell me i mean out of common sense i'm asking you of course we will be figuring it out a little later on if p then q if it is true does it mean if q then p will also be true do you think like that sometimes we're saying no right for example if i say if a number is divisible by nine then it will be divisible by three we all know it is a true statement okay any number which is will by nine that number will also be divisible by three but can i say the converse is true that means if a number is divisible by three it will be divisible by nine can i say that no i cannot because six or three itself they're not divisible by nine okay so please remember when your teachers in the junior classes used to prove converse of a statement separately they had a agenda they had a point for doing it right because the statement and its converse need not be the same okay anyways the next thing that you will keep reading is inverse of the statement inverse of the statement is read as if not p then not q okay this is called the inverse of the statement and there's something called contra positive of the statement contra positive is read like this if not q then not p okay we will see the tooth stables of all these in some time okay let me first ask you a simple very very simple question a very simple set of questions i have chosen for this read this question this says write down the contra positive and converse of these following statements okay the first statement says if it is a prime number then x is odd please tell me contra positive of this we will not do everything maybe we'll do first one second one and let's say fourth one yeah tell me the contra positive for this please write it down on your chat box today i will ask you to do some check typing activity so tell me the contra positive and tell me the converse contra positive is right there on your screen contra positive means if not q then not p by the way first in this statement if x is a prime then x is odd please note that p is x is a prime number and q is x is odd so what is contra positive of this correct contra positive is if not q that means if x is not odd then not p that means then x is not prime number it is not a prime number is it fine is it fine okay next one converse converse is this if q then p so how would you write the converse for this if x is odd right then x is a prime number simple nothing hard and fast okay we'll quickly do the second one also i think two of them is sufficient we don't have to do even the fourth one also so please complete the second one what is the contra positive here contra positive and converse if the two lines are parallel then they do not intersect in the same plane so how would you write the contra post if the two lines are parallel then they do not intersect in that same plane if the two lines intersect in the same plane two lines intersect in the same plane then they are parallel then the two lines are parallel anything you can write then they are parallel also i'm just saving some you know writing time that's why i'm not writing everything is it fine then then they're not parallel sorry sorry then they're not correct is it fine so if not q then not p okay what is the converse of this what are the converse of this if the two lines do not intersect in the same plane intersect in the same plane okay then they are parallel okay this is called the converse so any any doubt related to what is converse okay let me do one more example here the first one can you tell me the inverse also let's take one more inverse this is from my side even though the question is not asking you write down the inverse of the first one if x is a prime then x is odd what is the inverse of the statement writes at them if x is not prime then x is not odd or x is even okay whatever is this fine any questions now before i take a break i would take a break after giving you the truth table of if p then q okay so every kind of a compound statement we should know the truth table okay so let me go to the next screen truth tables so i will write the truth table for all of them and in the same go why to write it separately separately so converse inverse contra boster and we will analyze it also okay we will do the analysis part also so please pay attention everybody initially many people find this table to be slightly weird okay but let me justify the reason for the table okay so let me ask you the question first of all so i'm focusing first of all on this guy p implies q table okay since there are two component statements there will be four inputs t t t f t t f f correct now if p and q both are true by the way i forgot to tell you the name what we use for p and q let me go to the previous side i'll come back to it just give me a second p is called the antecedent okay okay sometimes also called hypothesis it's a name given to p q is called the consequent sometimes it is also called conclusion okay so here i'm making a truth table oh sorry i'm making a truth table for if p then q not you tell me everybody think and tell me if the cons antecedent and consequent both are true will that statement if p then q will it be true for example let's say i gave you a statement if it is a sunday then the school is closed okay now you woke up on a sunday that means it is a sunday you went to the school and you realize it is closed that means both antecedent and consequent are happening then will you call me and tell me sir your statement was true or will you call me and tell me sir your statement was false what will you call and tell me will definitely say sir your statement was true you were right so today's the sunday i came to school and i realized that the school is closed so whatever you were saying is true yes or no correct no but what if you realize that today is a sunday let's say on a sunday morning you went up and you realize that the school is functioning immediately you will ring me up and tell me what sir you were saying also true what will you say you're saying false isn't it yes or no why only one person is answering see i gave you a statement that if it is a sunday then the school is closed on a sunday let's say being a sunday that is true you went to school but the school is not closed so false means it is functioning it is open so you will call me and tell me what sir whatever you said that statement was true or that statement was false what will you say you'll say it's a false no because the school is functioning correct now you went on a non-sunday let's say monday you went to the school and the school was closed then will you call me and say you were true or will you call me and say you were false correct sathya correct arnav arnav you have already done this in school also correct this answer is actually true my statement still holds to be true because i said if it is a sunday then the school is closed you went on a monday that is no no no none of my problem it is closed open i don't care i mean my statement is not bound to you know become wrong because of that i said if it is a sunday if antecedent is not satisfied then i don't care about the consequent whether the school is open or the school is closed my statement still is valid my statement still holds so my statement is still true okay now if let's say it is a non-sunday let's say monday and the school is open does my statement become false because of this no it is still going to be true so please everybody note down this truth table because you will be using this very very frequently in solving the problem okay now many people ask me sir can you tell me a trick to remember this okay so normally i tell them sorry that let's say there's a husband okay who promises his wife who promises his wife that if i get a promotion okay if i get a promotion promotion then i will take you to singapore then i will take you to singapore okay for a holiday vacation okay now if this husband gets a promotion and he takes his wife basically it is the first case correct is he a good husband if yes it right or true if it is a bad if he's a bad husband right or false okay so first case is he gets a promotion he takes his wife to singapore is he a good husband of course yes he fulfilled his promise so true for this okay if he gets a promotion that means it is the first part becomes true the antecedent becomes true but he doesn't take his wife to singapore right he wants to save money and all so what kind of husband is he bad husband right so false isn't it if he doesn't get a promotion he still takes his wife to singapore that is your third part he's a very good person like despite he not getting promotion he still took his wife out to singapore for a vacation so he's a very good husband in fact isn't it yeah and the fourth case is he doesn't get promotion he did not get promotion he did not take his wife fine it doesn't make him a bad person right because really he doesn't have money i mean promotion comes with a lot of you know extra elimination and he did not get that so he did not you know take his wife to singapore so still is a good person right so this is why it is true is this fine any questions any question related to the truth stable now let us quickly complete the truth stable for the others as well so now i will draw the truth stable simultaneously for the converse for the inverse and for the contra positive now watch out the truth table very very carefully okay a lot of things we will take away from that see it now q becomes your antecedent p becomes your consequent okay so now read this statement in a reverse direction husband getting promotion taking his wife to singapore good husband yes did not get a promotion still took his wife to singapore good husband got a promotion did not take his wife finished right bad husband so it's false did not get a promotion did not take his wife still a good person so true now do you realize that these two tables are different from each other right so what is the conclusion here the conclusion here is p implies q so number one conclusion is p implies q is not logically equivalent to q implies p and that is the main reason why our teachers in our junior classes used to prove converse of a statement separately okay i personally i used to be very very agitated ma'am you are wasting time i'm doing the same thing reverse and all those things then i realize after doing this chapter that oh my teacher was right she used to prove the converse also because the statement and its converse need not be equivalent yes correct right i used to be very very agitated i used to tell my friends ma'am is wasting time and all this sometimes you realize you know what our elders do little later on in our life so when we keep saying no study hard study hard uts don't over indulge in ut and all you'll realize it later on anyways coming back sir you're giving philosophy also with logics next not p implies not q so let's let's make a table for not p and not q also quickly so that you know we save time can i make it here in this yeah let me make it so let's say not p not q just to save time because i will be needing it so this will be false false true true false true false true yeah so this becomes antecedent this becomes consequent yeah so husband didn't get a promotion didn't take his wife good husband didn't get a promotion took his wife very good husband got a promotion didn't take his wife bad husband got a promotion took his wife good husband okay now here you will realize one thing that these two truth tables are matching you realize that right so number two conclusion write down everybody number two conclusion converse is logically equivalent to the inverse okay converse is logically equivalent to the inverse correct please note it down very important now coming to the last one this becomes your antecedent this becomes your consequent so didn't get a promotion didn't take his wife good husband got a promotion didn't take his wife bad husband didn't get a promotion took his wife good husband got a promotion took his wife good husband okay now here what do you see these two are matching do you see that the yellow yellows are matching isn't it so what is conclusion number three conclusion number three is p implies q is equivalent to saying not q implies not p now here something very important see both the things actually are seen there's no difference it's just that you have flipped their names isn't it so if two is correct then three will be definitely be correct so this is not a surprise because just you have changed the name from p to q to p correct but what is important here to understand is that this is basically coming from the fact that q is the necessary condition so this is coming from so this entire thing that you are getting it's coming from the fact that q is the necessary condition for p which i already told you while i was explaining the topic to you correct that means if q doesn't happen p will also not happen see there getting on one so when you say if p then q then it is like saying if q doesn't happen p will not happen that means if the school is not closed it will not be a Sunday getting my point so that is the reason why i said q was the necessary condition now see truth table has justified it truth table has justified it that your consequent is a necessary condition for the accident got the point so this is something which is very very important when we come back after the break okay we will do a few questions on this concept and i'll also talk about if and only if and then few minutes we'll spend on validation of statements however for jee another exam validation is not important but maybe for your school they can ask you so right now we'll go for a break time now is 6 22 let's meet exactly at 6 37 p.m okay 15 minutes break eat something have something some food and we will meet after 15 minutes okay enjoy your break students ask me this question can you write if p then q in terms of your and not or your or operators that you have already used okay so here is a you know actually a question for all of you prove that if p then q is logically equivalent to not p or q right so you can generate what i'm trying to claim here is that or what i'm asking you to prove that the truth table for this can also be obtained by doing this operation on these two statements okay so if p then q is logically equivalent to not p or q please prove this by using truth stable everybody please do this and tell me are you able to prove this or not i'm not solving it i'm just making my tables so that i can use them p q not p make the truth stable everybody i'm giving you one minute time and let me know are you able to prove it just say yes if you're able to prove it no if you're not able to prove it okay done achintya very good so achintya is the only person who got it or anybody else got it satyam also got it well done neel omesh satyajit smithi venkat vishal yuvan harsada without waste but much waste of time let's do it so this was our four entries for the two component statements negation of p will be false false true this you already know from our previous okay exercise now negation p or q so true or false or false or true whatever you want to call it that will be true only false or false will be false true or true will be true false or true will be true so what do you realize you realize that it has the same truth stable correct which means yes these two statements are logically equivalent okay so your if p then q can actually be written in terms of or and negation operator like this okay now from here very very important takeaway is this that if you want to negate so important to note that if you want to negate if p then q it is equivalent to saying negation of negation p correct and as per your duality or you can say as per your de morgan's law this will become negation of negation p or will become and negation of q so please note this down everyone negation of p implies q negation of p implies q is logically equivalent to saying p and not q please note this down any questions any questions all right so let's take few more questions where implications are involved we will not take many questions by the way okay let's take this question this could be a multiple option correct question okay so please give me a response on the chat box so the question says p q r are simple propositions p q r are simple propositions with truth values of true false and true respectively okay then what is the truth value of negation p or q and negation r implies p so this is a symbol for implies the same symbol so instead of putting a single arrow they have put a double arrow doesn't make a difference so more than one option may be correct here maybe i'm not guaranteeing that it's always have to be correct but maybe correct so give me all the responses that you feel are correct here on the chat box okay sattim so you feel only one option is right okay well the question is slightly written in a confusing way in option c and d they have all they have changed r to false so earlier in the initial question they gave r as true then they change r to false so let's say if it is true false false then this statement will be true that is what they're trying to say and in option d they have changed q to true so if it is true true true okay then the statement is true that is what they're trying to say oh i guess that that you would have you know miss misread or misinterpreted the question so c and d question sorry c and d option basically says that if you change only that r to false then that statement will become true okay and if you change your q to true then the statement will become true that is what your c and d option say okay so p q r negation p negation r we need negation p or q we need negation p or q and negation r and then final statement which is okay we shall yes ready everybody okay satyam okay so what initially they have given is that this is true this is false and this is true okay so this becomes false this also becomes false okay now negation p or q so false or false will be false okay false and anything will be false by default okay nothing this is your antecedent and this is your consequent so husband not getting promotion taking the wife to singapore good husband so true so as per this as per the initial condition given to us of true false true this is true so option a is for sure correct okay next in option c so b cannot be correct we shall sorry b is wrong then option c they have said make r as false so let this be true this will be false and make r also as false so let's see what happens so this becomes true okay this becomes again a true okay negation p or q negation p or q will become a true okay and negation p or q and with negation r that will also become true if i'm not mistaken because negation r is also true so these two and will also be true okay now husband getting a promotion taking his wife to singapore yes he's a good husband so this will be true so option c is also correct is it fine any questions related to option c being correct okay now let's take option number d d says make q as true then see what happens okay let's see so p is true q also make it true and r they have given already in the initial part as true okay so r will be by default taken as true so this will become false this will become false now negation p or q negation p or q will become false negation p or q will become false false and anything will be false okay now husband getting not getting a promotion still taking his wife to singapore trip he's a good husband very good husband so t will be true will be correct for d as well so a c d a c d is going to be right yes at the many issues let me know didn't get how c p is f no p is to this as per the question p is always true why should i put f sorry negation of p is f okay so f or q okay f or q this will be f oh sorry sorry f or q will be f here f or q will be f here okay and f intersection this will be f correct but thankfully this doesn't change my answer satyam clear thanks for highlighting that out but thankfully it did not change my answer thankfully achintya what was your question are you not clear how did i get c option correct that's what you were doing right now got it okay now coming to the second implication which is your if and only if implication so b is if and only if so if you have a statement like if and only if p then q it is basically read as p is the sufficient and a necessary condition of q and vice versa q is the sufficient and necessary condition for p i think i should use for here okay that means both implies the other that is why we use the symbol with a double arrow like this okay now logically speaking i mean logically speaking if somebody says if and only if p then q then these two statement actually become equivalent that means if somebody says p automatically q will imply if somebody says q automatically p will imply so they will become like the jv of sholay movie okay so if p is happening q will happen and if p is not happening q will also not happen are you getting my point so one will imply the other so we say normally that this is logically equivalent to saying p implies q and q implies p okay so when somebody says if and only if p then q he indirectly means p implies q and q implies p that means p implies and is implied by q okay now let me explain this so a simple example so if i give the statement if and only if it is a sunday then the school is closed then the school is closed okay then the school is closed so if this statement is correct or if this statement is true then it means that if somebody comes and tells me hey it's a sunday then it automatically implies the school is closed and if somebody comes and says hey if it is not a sunday then it automatically implies that the school is open are you getting my point and vice-versa also so if somebody comes and tells me hey the school is closed that means automatically it is a sunday or if somebody comes and tells me hey the school is not closed then it is automatically not a sunday so they will both occur together or they will not both occur they will both not occur together so there's many close friends right so p and q will together be true or together will be false then only this statement will be true or if this statement is true both have to be true or both have to be false are you getting my point what i'm trying to say right so one implies the other that is what we need to understand here so any one statement of the two if i say other will automatically be applicable right so in light of this the truth table that we normally made for if and only if p then q is something like this please note this down true will be only happening in these two cases when both are true or both are false okay if any one of them is true another is false it will become false okay that means let's say i'll justify this truth table if i give you this statement that hey only if or if and only if it is a sunday then the school is closed correct now let us say it was a sunday right you went to school your school was closed will you call me up and say you were true or will you call me up and say you were false you'll definitely say you're true sir okay but if you go to the school on a sunday and you realize the school is working then obviously my statement becomes false right sir you're told if and only if it is a sunday the school is closed i went on a sunday school is functioning so you're false correct and if you didn't go on a sunday let's say you went on a monday and the school is closed then also my statement will be false because i said the school will be closed if and only if it is a sunday so how come the school is closed when it is a non-sunday correct then also my statement will become a false but if you don't go on a sunday and the school is open then my statement still is valid because i said if and only if it is a sunday then the school will be closed did you go on a sunday no is the school working is the school working yes so if it is if you don't if you're not going on a sunday the school is working that is none of my problem because i said if and only if it is a sunday then the school will be closed are you getting my point so my statement still holds valid so please understand here in the truth table only when both are true it will be true or only when both are false it will be true and as in all the cases it will be false false so as a homework please verify this by using truth table okay i you already know your uh truth tables for implications sorry conditional implications if then implications so verify it by truth table in your as your homework is it fine any questions any concerns okay now once we have learned this the negation is also very simple to write down so negation of please note this down negation of negation of if and only if p then q is equivalent to the negation of this guy p implies q and q implies p correct which is equivalent to i hope i have space here yeah which is equivalent to saying negation of this or negation of this okay which is equivalent to saying no now the negation of this we have already seen in the previous slide p and a not q or negation of this will be q and a not p is it fine sorry i should say note it down oh i already written it so please make a note of this very very important any questions let's take a small question uh this question came in Manipal exam okay Manipal takes its own entrance exam if p and q are two statements then negation of p and q or negation of if and only if q then p okay is a tautology contradiction neither tautology nor contradiction either tautology or contradiction let's do this question i'm launching the poll for all of you please whenever such question comes make good use of the truth stable make good use of the truth stable those the person who has answered with d d is definitely not going to be the answer so i would request people who are answering now onwards either tautology or a contradiction it cannot be both so it's a statement so you're stating the statement is always false or it could always be true right it cannot happen like that it has to be one of them if at all it is one of them so d is so either it is tautology or it is a contradiction or it is a contingency so these are three you know disjoint sets it'll fall in either one of the category okay sattim okay i can give you 30 more seconds so please complete this two table doesn't doesn't take much time i think one one and a half minutes is good enough to make a table with two component statements so only two component statements i can see in this question p and q should not take much time okay five four one okay end of poll out of seven people who have responded four of you have said option number c all right so let us make a truth table for the same so here the two table required is of p q p and q negation of p and q if and only if q then p negation of that and the final w f f okay so this is p this is q this is p and q this is negation of p and q this is q okay if and only if q then p then negation of that guy also and then final let me make it properly yeah and then final statement which is this very straightforward this is not a rocket science okay i hope everybody is finding it comfortable to understand so true true true false false true false false p and q uh p and q will be true false false false negation will be false true true true now if and only if q then p only when both are true it will be true or only when both of them are false it will be false so it will be like this okay negation of this will be false true true false now this and this you have to take the or so false or false will be false the moment you get a true here stop in fact let me complete it so it is neither tautology nor a contradiction so it is actually falling in option number c it's a it's a contingency is it fine any questions any concerns any questions any concerns so we are not going on to the last part of the chapter which is basically nothing nothing but validation of validation of implications validation of implications so if you have been given an implication how do you validate it that is what we are going to talk about it okay just give me a second yeah sorry somebody was at the door yeah now a validation of implication how do you validate an implication so basically if i give you a statement if p then q okay how do you validate the statement so to validate the statement there are three ways to do it or there are three you can see methods that we normally follow one method is called the direct method okay so we'll discuss this method the second method is called the contra positive method and the third method is known as the contradiction method okay now how do these methods work let me explain this one by one to you in fact i will give a take an example also to support it so the first method direct method how do we prove that how do we prove that a given implication if an if p then q is true by direct method we follow the following three steps step number a we assume that p is true okay so what do we do we assume that this antecedent p this is true and using this assumption using this assumption that means using a we prove that q is true that means if both p and q are true we know from our truth table we know from our truth table that p implies q will also become true remember the truth table husband getting promotion and taking his wife good husband so true correct so how does this method work with an example let me explain you so if these two are done that means this implies that this is going to be true let's take a small question and i will explain it via that question so how does direct method help you to prove this so let's say i want to prove this statement so there's a statement q sorry there's a statement p this says that if x is real number such that x q plus 4x is 0 then x is equal to 0 okay we need to validate the statement we need to validate means we need to prove that the statement is true by using direct method so i'll be only talking about direct method because i have not done contradiction method and contra postive method yet so this is how we approach so first of all i would break p as something like r implying q okay or r implying s let's take s so please note that the name has got changed okay they have started calling okay you do one thing you start calling this as r itself so that you know we continue using our old name p and q so just so that you can match whatever i have written in the theory part with this okay so let's say this is your statement r and you want to validate this r statement so what do we have done we have written it as a p implies q statement where p says x is real and x q plus 4x is 0 okay and q says x is equal to 0 so you want to prove this statement by using direct method so what do we do we first say let p be true okay that means we are assuming that x q plus 4x is equal to 0 and x is a real number now this will imply that this will be true correct and this can mean only two things either x could be 0 or x square plus 4 could be 0 but this is not possible why this is not possible because x is real number right you can't have a real number square plus 4 equal to 0 so this finally implies that x is 0 right so if x is 0 means you have proven that q is true correct so what did i what did the direct method say assuming p is true if you prove that q is true then if p then q will become true that means our statement that was given to us is actually a true statement is it fine any questions is it fine any any question related to how does a direct method work clear now these parts will not be tested in your competitive exam so this is only for your school exam i'm telling you if at all they come in school exam but many a times in class 11th the teacher is the decision maker if she feels this part is not going to be useful in 12s she might not even cover it it is up to her okay so when you do that chapter in school you would realize that it is actually used you know required for you or not but of course it is not there for competitive exam okay competitive exam is not going to ask you to prove something the second method is the contra positive method the contra positive method basically works on this principle that this statement is logically equivalent to this statement correct you already know that if p then q basically is equivalent to saying if not q then not p because q is the necessary condition for p isn't it so what do we do in this method we apply a direct method on this game that means we assume that negation q is true correct that means you are assuming that q is false okay and then by using this assumption right we prove that negation p is true that means you indirectly prove that your q is false sorry p is false so what are you doing you are actually applying a direct method to this game so in direct method what did we do antecedent we used to assume it is true and then the consequent we used to prove that it is going to be true because of that assumption that is what we did in the previous slide so here your antecedent is negation q so you are assuming that your negation q is true and using that negation you are trying to prove that negation p is true so when you do that then by direct method basically this will become true correct and since these two are logically equivalent this will also become true right so you are coming from your contra positive direction to prove the validity of the statement and that is why the name of this method is called the contra positive method okay let me demonstrate this let me demonstrate it in the same example which i took a little while ago but make a note of this first of all learn copied okay let's take the same example so the same question now i'm going to solve by using the contra positive method which is your third method away i'll come to contradiction after this okay so here i will start with again let me name this guy to be r first of all so let's call it as r okay let's call this part as your p and let's call this part as your q okay so here we'll start with the fact that let q be false that means i'm doing step number one see here step number one negation q is true means q is false that means you're trying to first start with let x be not equal to zero now we already know that x square plus four is going to be a positive quantity if your x is real so if you multiply this with the first one so let's say i call this a second one this side call it as the first one second one now if i multiply a one with two so basically you're trying to say that this term will be not equal to zero because anyways this is positive and this is not equal to zero and if you're multiplying a not equal to zero with positive it will definitely be not equal to zero okay in other words you're trying to say x q plus four x is not equal to zero what does it mean you're trying to say that p is false correct so basically what did you achieve here you first of all started with the assumption that negation q is true and you ended up proving that negation p is true so in light of the direct method you have actually proven that this is true and by the fact that this statement is the logical equivalence of this you have actually proved that this is true done is it clear so this whole thing is basically based out on the fact that q is the necessary condition for p so you first prove that the contra positive of that statement is true somehow by the use of direct method and because the contra positive is true the contra positive is the logical equivalence of if p then q so both will become true then is it clear any questions any questions any concerns coming to the third method so we have already done direct method contra positive method now the third method is called the contra addiction method see contra addiction method is not a new you know method for you all maybe you did not know the name for it but you were already using it in class 10 i'm sure most of you would remember this question prove that root 2 is a rash irrational number did you solve these kind of questions in class 10 i'm sure in abundance you would have solved it in abundance root 2 root 5 root 3 any any type of thing root 7 anything would be made as a question yes correct so actually you use contra addiction method there so you started with something which is like contradictory to what you are supposed to prove isn't it so let's say you were asked to prove that prove that root 2 is a irrational number so you started with let root 2 be a rational number and then you did something something something and you result and at a step you came into a contra addiction so you came to a fact that which is not possible it is contra addiction contra addiction means not possible and then what you did you basically changed your assumption itself by the law of extruded middle so if your root 2 is not a rational number it has to be irrational that is how you used to prove it so more or less the similar approach will be utilized here also so what do we do here we basically follow the truth table where it said that where it said that if husband got a promotion and wife was not taken to Singapore or he did not take his wife to Singapore he is not a good husband so the only situation so basically this method is based out of this truth table so the fact that antecedent is true and consequent is false that only makes sure if a if then statement false that is going to be used over here so what I'm going to do here is if I want to prove that this is true if I want to prove that this is true what I will do I will do this I will say let P be true or I will assume P to be true and Q to be false indirectly what am I doing so instead of proving to I'm assuming that okay let P be true and Q be false means I'm assuming that this is false isn't it so this is indirectly what I'm assuming am I right see what I have to prove I have to prove root 2 is rational sorry irrational so I'm assuming that let 2 root 2 be irrational oh sorry what I'm saying if I have to prove that root 2 is irrational I'm assuming that let root 2 be rational okay so all time assuming the opposite I'm assuming so for assuming that this is false I have to assume that P is true and Q is false and with this assumption you actually reach are you end up are you reach a contradiction reach a contradiction means you reach a step where you realize that that you know that is not possible that is always a contradiction it cannot happen okay so in this case what do you do you change this assumption so this assumption becomes wrong the assumption that so this implies that the assumption of P implies Q is false is itself false okay so your assumption that it was false becomes false so by convolutional law what does it mean this is true so this is another way of proving it okay let me demonstrate it then you will understand what do I mean to say first note this step down then I will demonstrate these steps in the very same example tomorrow again 4 to 7 30 we'll have a class on parabola it's not going to be a lengthy session one class is good enough to complete parabola as well okay so tomorrow's class will be end of parabola and one more thing is the next week we will still have two classes Friday Saturday but post that week we will only have one classes as usual and that too on Tuesday so we'll go back to the Tuesday schedule okay so next week we will have Friday Saturday once again and next to next week we will have only on Tuesdays so you'll get back to your Tuesdays classes okay anyways let's solve this question so I'll be now solving this question by method of contradiction so again let me call this as okay and let me call this as P and let me call this as Q so in contradiction method what do we say we say let P be true and and Q be false so basically you have used the AND connector here so what you're doing you're assuming P be true and Q be false that means you're saying X Q plus 4 X is equal to 0 okay and at the same time you're saying X is not equal to 0 okay that means you're saying X into X square plus 4 is 0 and at the same time you're saying X is not equal to 0 correct that means you're trying to say X is equal to 0 because your X square plus 4 cannot be 0 okay so if X is a real number if X is a real number then X square plus 4 cannot be 0 right so this automatically implies that only X could be 0 so now you're writing something like this you're saying X is 0 and X is not 0 also isn't this a clear cut contradiction you're saying P and so you're saying Q and a not Q Q and not Q is a contradiction isn't it they cannot happen together okay this is clearly a contradiction step correct so that means this assumption let's say I call this assumption as assumption A okay this assumption will become a wrong assumption right so this clearly states that your assumption A assumption A is false right that means you're trying to claim that P is 2 and Q is false itself is false right that means you're trying to claim that this is false so what does this mean this means it is false right so assumption A is trying to say that this is false so if assumption A itself is false means this is true simple as that so this is how you prove it by contradiction so thanks to the fact that a statement cannot be true and false simultaneously that's the whole beginning point of our you know discussion remember I started my discussion by saying that a statement cannot be true and false together so that thing is actually saving me here because if a statement could be true and false together then I could not use this method of contradiction to solve it okay so this is how you validate it now when it comes to if and only if for if and only if I don't have to give you a separate gyan because if and only if is okay so let's talk about validation of if and only if see if and only if is basically made up of this and this correct now if you want to validate it use any of the methods to show that both have to be true because if any one of them become false you know that a conjunction will become a false isn't it right so in order to prove that this is true you have to show that this is also true and this is also true and for this you can use any of the three methods that you feel like okay you could feel you could use method of like you use direct method okay or you could use contradiction method or you could use contra positive method okay any of the three methods could be used here okay so for proving this you can use for these two you can use this so basically you have to validate these two separately by any of the three methods so this is not you know completely different from what we already did it is just doing it twice okay so you have to prove p implies q is true you can use any of the three methods and you have to also prove that q implies p is true you can use any of the three methods okay so once both are true then only you can say if and only if p then q is a true statement okay so there's no point taking any question here because it is just the same thing doing same you know type of methods using use twice okay so this is how you know you will be asked questions if at all you are asked these kind of questions in your schooling set so we cover up this topic as well i will be sending you one worksheet based on the same okay thanks a lot tomorrow see you again i will send you the reminder right now so see you tomorrow with parabola session yes this chapter is over okay there's nothing in this chapter other than these type of compound statements okay so you can solve all the j problems based on whatever we have done all right bye take care everybody