mathematical reasoning just like statistics mathematical reasoning is a very sweet and simple chapter and I think one question definitely comes on it mathematical reasoning this chapter is based on basically the concept of logic okay or the concept of deduction okay so in logic there are two aspects of validating a statement one is by induction another is by deduction induction is where from given instances you are trying to generalize things and deduction is where from generic things you are trying to you know find an answer to a very specific case okay so there are two parts of reasoning one is called induction other is called deduction induction is what you studied in your principle of mathematical induction and deduction is what you are going to study in this chapter which is mathematical reasoning okay so in this chapter we are going to first of all learn what are statements okay what are statements what are the different types of statements let me write them down so the overview of the chapter is overview of the chapter is we are going to talk about statements okay second is types of statements they are truth tables type of statements we are going to talk about compound statement of a simple statement is also there compound statement if then statement if and only if statement okay along with the truth tables of these and finally we are going to learn about validation of statements particularly validation of the last two if then and if and only if okay so this is what we have in this chapter not a very difficult one very scoring mostly this is the important part of this chapter okay so let's begin let's begin first of all statements I think I have already told you before the difference between a sentence and a statement sentence versus a statement so what are the difference between a sentence and a statement sentence is basically a collection of words which make a certain meaning okay and sentence could be of four types it could be a declarative sentence which we call as sometimes assertive sentence okay sometimes you also call it as the proposition where you try to declare something right for example 3 is a prime number okay so this is a declaration which I'm making if this is an assertion which I'm making this is a proposition which I'm making or let's say son is a planet okay that's what my daughter's teacher says son is a planet moon is a planet I don't know for what reason they say so but I that's how very small kids have been taught son is a planet okay now these are declarative sentence now I may not be correct for example son is not a planet son is a star if my geography is not that weak son is a star okay so when I say something it is just an assertion I may be wrong in my assertion okay second type of sentence is imperative sentence imperative sentence imperative sentence is where I'm making some requests or giving some command to somebody like switch on the lights or switch on the fan okay or please please give me a glass of water or give me a cup of tea okay so here I'm making some kind of a command or a request okay so those are called imperative sentence means I'm trying to put a force on the person who is basically trying to listen to these commands third is interrogative sentence now my purpose is not to teach you English here because many times in the J exam they ask you which of the following is a statement okay interrogative is where you're trying to question somebody okay so how was your math paper how was your maths exam so I'm asking you a question okay why you did not appear for the last came on test so these are interrogative sentence the fourth one is an exclamatory sentence exclamatory sentence is where some kind of an exclamation is used where you're trying to express some kind of an excitement sorrow joy okay like let's say wish you a happy birthday wish you a happy or have a good day have a good day okay so these are exclamatory sentence now out of the sentences those sentence those sentences which you are those sentences which are first of all unambiguous unambiguous means there is no doubt about it that means there is no element of uncertainty about it and those which can answered we those which can be answered with a true or a false but not both so those statements which are unambiguous and can be answered with true or a false but not both are eligible to be called as are eligible to be called as statements so statements are those type of sentences which are unambiguous and which can be responded with the true or a false but not both if you are responding any statement with both two and false it actually becomes a paradox okay so paradox is something which is it can be both true and false let me write both true and false okay so statement are those sentences which are unambiguous and which can be answered with a true or a false but not both so out of these four types of broader division of sentences which of them can be called as statements can I call an exclamatory sentence as a statement if I say have a good day will you answer this as a true or a false will you say true I don't think so you say thank you okay and wishing is the same right it cannot be true or a false I am nobody to decide whether you when I say you have a good day you will have a good day you may have a bad day as okay interrogate a sentence can I call it as a statement I don't think so if I say how was your maths exam you will say false I don't think so you'll reply with a false you say it went well or it didn't go well or it didn't go as expected okay so exclamatory sentence cannot be a statement interrogative sentence cannot be a statement imperative sentence if I say switch on the fans will you say true will you say false you say yes I will or no I won't right please give me a cup of tea will you say true you'll say okay yes I'll get you a cup of tea or no I will not get so this cannot also be answered with a true and a false but if I say a declarative sentence or a search this sentence or a proposition that's a three is a prime number you'll say true yes whatever you're saying is true okay son is a planet false so these type of state sentences will be called as a statement so this is only statement possible okay statements are normally in mathematics written by a small alphabet P Q R whatever you want to say so let's say if I say son is a star sorry son is a planet by the way I I'll resume the same statement which I said in the previous slide is son a star is a interrogative sentence it is not a statement guy 3 you're not asserting something you're asking something no it's a question it's an interrogative sentence so interrogative sentence are not statements and is son is a star use don't say true for it you see yes for it or no for it get the difference meaning yes no and true false are you getting my point so when you say true it's a fact when you say false right it is not it's not a correct statement it's a false statement but yes or no is something like you're you know you are willing to obey somebody or you're willing to answer somebody okay when you say yes do you think you're going to yes is a correct thing for it it could be maybe also right so when you ask the question it is not a statement when you assert something when you declare something when you propose something that is a statement so this is a basically a statement which I have denoted by a small alphabet P okay you can use any alphabet not normally P is used P for proposition and there's something very interesting negation negation of a statement is written by a till the sign attached in front of it okay I have seen P dash also been used or P bar also been used but mostly you'll see negation symbol being used negation is nothing but it is the opposite or you can say let's say son is a planet so negation of it is son is not a planet okay simple as that many a times some students write it is not the case it is not the case that son is a planet okay keep it simple now when it comes to negation there are certain questions which will be having quantifiers into it now what is the quantifier let me just talk about it there are two types of quantifiers one is called a universal quantifier another is called a existential quantifier existential quantifier what is a universal quantifier all the statements which have got words like or phrases like all every okay or none okay so statements which begin with or have these kind of wordings in them they are called universal quantifier that means universal means all the objects come into it so they quantify the entire object for example if I say all cats scratch so basically giving a universal quantifier to this statement where I am including every cat existing on the planet or for that matter universe okay so I'm saying every cat in this universe will scratch so this is a unit statement having a universal quantifier there are some quantifiers which have state which have words like there exists I'm sure you would have heard this in maths also okay there exists or at least one okay or some so whenever a statement contains these kind of words or phrases we say that the statement has got existential quantifier that means there is an existence of certain objects which will satisfy that characteristic for example if I say there exists a prime number which is even so yes we know that to is a prime number which is even okay or there exists there exists a person in this planet who is 8 feet 6 inches tall okay or some people in Karnataka speak Hindi so these all statements are containing existential quantifier so first of all is it clear what is universal quantifier and what is existential quantifier now when the statements begin with these quantifiers and when you're finding out its negation these be very very careful for example let me ask you this question let's say I give you a statement R this says all cats scratch can you give me the negation of it I would love I would like everybody to give me a response on the chat box all cats scratch okay right now many of you have given a wrong answer saying that all cats don't scratch or no cat scratches etc. please note that they're all wrong the answer to this is at least one cat doesn't scratch or there could be various versions of it there exists a cat that doesn't scratch yeah that is also correct some people have said not all cats catch that is also fine but normally we do not begin a statement with not all okay better to say some cats do not scratch right so guys this is a mistake which people do when your statement begins with a universal quantifier your negation should begin with a existential quantifier get this correct so when you say all cats do not scratch it is covered under this so when you say all cats do not scratch or no cat scratch they are all covered under these okay so when I say all cats scratch the negation of it is getting to me at least one cat which doesn't scratch for example if I give a statement hey all cats scratch tomorrow you come to me with a cat in your hand and say sir see this cat okay no matter how much you you know tease it it will ever scratch you and of course I realize it doesn't scratch so what I said will become wrong or whatever I whatever I have said will become false so even one instance you are able to show me where whatever I have said is going to fail that means that statement is getting negated are you getting my point so you don't have to show me all the cats in the world right and show me sir see these all cats in the world don't scratch that is fine I mean that is going to negate my statement but even if one cat doesn't scratch my statement gets nullified my statement gets becomes false invalidated are you getting my point so here is a small concept for you if a statement begins with a universal quantifier its negation will be having existential quantifier and vice versa let's say had I given you a statement that there exists a cat that doesn't scratch then its negation would be all cat scratch get the getting the point so please make a note of this this is something which has been asked in J before and let me pick up a question based on it I think I thought what to take a question but just give me a minute I'll pull the one out okay so the question goes like this just a second I have I think statistics statistics okay I don't want to write actually that's why I'm pulling out this question where is my question okay there you go I hope you can see this question I'll do one thing I'll just take a snapshot of this read this question carefully SB a non-empty subset of R consider the following statement P P says there is a rational number X which comes from this set S such that X is greater than 0 okay which of the following statement is the negation of this statement I'll put the poll on let's have one one and a half minutes I think one minute is good enough think carefully and then answer think carefully and then answer okay let's stop the poll I think 11 of you have voted 5 4 3 2 1 go okay most of you have said option number B which is actually correct okay so see when you have there is a rational number basically there is a rational number shows existential quantifier okay this is an existential quantifier so when you're asking for the negation of an existential quantifier it must have a universal quantifier so there is no is a possibility and every is a possibility both are universal quantifier none or every they are all universal quantifiers now when you say there exists a rational number which is greater than 0 then of course you can say there is or you can say every rational number is actually less than equal to 0 and option number C becomes the right option are you getting my point next moving on to compound statements what are the compound statement compound statement is basically a type of a statement which involves simple statements so we are going to learn four types of compound statements in our today's session the first is a compound statement where two statements are connected by and operator or and connector which we call as conjunctions okay what is the conjunction conjunction is basically when a simple statement P and a simple statement Q are connected by and operator okay symbolically we represent it as P a paro Q this is the same and operator that you have actually learned in your Boolean algebra and computer science okay so some examples I can give for this let's say 18 is divisible by 3 and 6 so if you see this is made up of two simple statements connected by and operator so what are the simple statements here so let's say I call this as R so you can have one simple statement is 18 is divisible by 3 and another simple statement is 18 is divisible by 6 so when you connect these two statements when you connect these two statements by a and operator it becomes your statement R so now when you are trying to compare two statements so we say they are logically equivalent okay so this is a simple which is normally used when you are trying to show that two statements are logically equivalent what is logically equivalent logically equivalent means they mean the same thing actually or mathematically speaking they have the same truth table okay so when two statement have same truth table they are called logically equivalent okay we'll talk about it in some time but however I just wanted here to make you understand that this is a compound statement and these are two simple statements which are connected by an and connector and these kind of statements are called conjunctions these type of statements are called conjunctions now let us talk about the truth table of conjunctions most of you already know that because you have done this in your junior classes or in some computer science course so conjunctions are true only when both the input statements are true if any one of the statement becomes false conjunction becomes a false okay by the way if there are two statements involved in a conjunction please note that there can be conjunctions between several simple statements so you know that if there are two statements involved then two to the power two inputs should be given which is four one two three four if there were let's say three statements PQR connected by let's say and operator or and connector then eight inputs will come true true true true true true false true false true false true true then true false false false false true false true false false false false like that if eight combinations will come okay so this is the truth table of a conjunction any question about it this is already known to most of you so I will not waste much of your time the next compound statement that we are going to talk about is a disjunction disjunction disjunctions are those statements where simple of statements are connected by or operator or or connector this or is an inclusive or this or is an inclusive or what are the meaning of inclusive or inclusive or means even if both of them can happen together so both P and Q can happen together also for example if I say let's say you're going out to buy something okay so I say okay get me get me let's say pastry or cake get me pastry or get me cake that means even if you get both for me I'll be happily accepting it okay I'll be more than happy and I'm getting two things to it but if I say something like this the children are in the classroom or in the playground this or is an exclusive or because the children cannot be simultaneously in the classroom and in the playground okay they can only be at one of the places so this is an example of exclusive or so what are we dealing here is with is inclusive or so we are talking about inclusive or where both P and Q can happen together okay symbolically represented by this symbol so example as I have already said so I'll give you some mathematical example let's say 3 is an odd number or a prime number okay so it is made up of disjunction of two simple statement 3 is an odd number or 3 is a prime number okay simple as that nobody has any problem in understanding it let's look into the truth table of it so truth table for a disjunction looks like this so at least one of them has to be true for the statement to be true so true true is true true false is true false true is true but false false is false so at least one of them has to be true for the statement to be true okay now coming to the negations of these two statements which is very important so negations of conjunction and disjunction negation of a conjunction and a disjunction follows the de Morgan's law so if I say negation of P and Q it is logically equivalent to negation P or negation Q right very much similar to what we had done in our sets A intersection B complement is A complement union B complement okay so basically de Morgan's law is applicable to this as well negation of P or Q is logically equivalent to negation of P and negation of Q please make a note of this very very important these two are called the de Morgan's law okay now I'll just show you the proof for one of them I will draw the truth table let's say I draw the first one I'll draw the truth table for the first one and show you that the truth table of this will match with the truth table of this so let me just yeah so P Q negation P negation Q P and Q negation P and Q I think I need more and finally negation P or negation Q now just watch out for the outcomes in these two columns you will see that they will have the same truth table so let's start true true true false false true false false so these are the four combinations of the input statement negation P means true negation is false false true true negation Q will be false true false true P and Q will be true false false false negation of this will be false true true true now negation of P or negation of Q so you just have to or it so false or false is false false or true is true true or false is true true or true is true as you can see they have the same set of outcomes that means they have the same truth table same truth table means these two are logically equivalent so they are same so they are logically equivalent that's why the symbol here I have used logically equivalent is it fine so many a times a question will come to prove whether two statements are logically equivalent or not or which of the statement here is logically equivalent to the given statement so please make truth table for those cases don't worry we'll be taking questions on that we'll be definitely taking questions on that is it fine any questions here any questions any concerns okay so before I move on a quick word about the truth table in some cases you would realize that the truth table always comes out to be true true true for a given statement okay a simple example of this could be P or not P okay if I make a truth table for this let's check it out P not P P or not P so let's say true and this will be false and this is false this will be true correct so if you take their or it will always be true true so you realize that the truth table is always containing true true in such cases such statements will be called as a tautology so what's the tautology tautologies are those statements whose truth table is always true that means those statements are always true they can never be false get the getting the point on the same hand if you have a statement whose truth table is always false for example rhyming with the same P and not P if you see P and not P I don't want to make it again I'll just write it over here so if you see P and not P will always be false okay so both answers are coming out to be false so such statements are called fallacy fallacy fallacy contradiction many words are given to it okay okay fallacy or contradiction okay and statements which are neither tautology nor fallacy statements which are which are neither tautology nor fallacy I think I need to switch on my camera or switch off my camera it's it's becoming slow yeah fallacy it is called a contingency okay contingency so contingencies are those which have got both true false outputs coming out in the truth table so direct questions have been framed in JEE where they will ask you is this following or which of the following statements is a tautology okay or which of the following statement is a contradiction okay we'll take examples don't worry because I have few more cases to cover up and then we'll go on to some examples so as of now under compound statements we have covered conjunction truth table of conjunction disjunction truth table of disjunction and negation of conjunction and disjunction is that part clear to everybody any questions I hope you're finding this chapter easy interesting and it's very scoring by the way nobody ever misses out solving a question on uh mathematical reasoning in the JEE main exam so if you feel you do not like this chapter then you are missing out on a sure shot four marker in your JEE main exam okay next coming to the main important ingredient of this chapter which is the third compound statement called if then statement if then statement if then statement is also called implication by it's also called uni directional implication uni directional implication okay now why it is called uni direction implication you will come to know from the definition only so when we say if p then q so this is a typical example of an if then statement or this is the structure of an if that statement right so let's give me let's take an example uh let's say if you study hard hard then you will qualify JEE okay so here this has two simple statements so one is if you study hard so you study hard is a simple statement you will qualify JEE is another simple statement okay so when two two statements like this are connected by if then statement this is called a uni directional implication so a single arrow is used here right because if this happens then this will happen not the other way around okay by the way p is named as p is called as the antecedent okay antecedent also called as the hypothesis q is called the consequent also called as the conclusion okay so when you say p single arrow q in English language we read it as if p then q okay p is called the antecedent antecedent means it'll occur before consequent is it is it'll occur as a consequence of it or it will be an implication of it so you studying hard your implication will be you will qualify JEE are you getting my point here so I hope it is clear what is if then and how it is represented okay by the way there are three associated terms associated with it one is called converse of this converse of this is written as q implies p so when you switch the position of the antecedent and consequent it is called the converse of this statement by the way they are not the same thing right they're not the same thing how it is different why it is different we'll come to know from their truth tables don't worry about it second thing is inverse of if p then q okay this is written as if not p then not q this is the inverse of if p then q and the third is which we'll get to hear very frequently is contra positive of if p then q and that is written as if not q then not p please note this down so this is read as if p then q okay and this is read as converse of if p if p then q this is read as inverse of if p then q and this is read as contra positive of if p then q any doubt any concerns any doubt any concerns please do let me know right now because this is one of the most important ingredients of this chapter almost 80 percent of the questions that you will see will be basically testing you on this okay now let us look into the truth table yes I can see some questions coming up could you explain it cause I got disconnected for which part you want me to explain Shadda if p then q this is called converse of if p then q this is called inverse of if p then q and this is called the contra positive of if p then q okay p is called the antecedent q is called the consequent okay so if the consequent and the antecedent positions are interchanged you get the converse of that statement okay if antecedent and consequent are both negated you get the inverse of that statement and if your antecedent and consequent are swapped and negated you get the contra positive of that statement by the way very interesting observation that you can see here is that the inverse of this is actually the contra positive or you can say yeah contra positive of the converse okay so this is the contra positive of this statement okay inverse of if p then q is the contra positive of the converse of if p if q then p getting my point so see here what is the contra positive of this contra positive of this is if not p then not q and which is actually the inverse of this okay so both these statements are logically equivalent in fact their same statement has to be logically equivalent okay so let me write it they're actually the same in fact same okay we'll talk about all these things with respect to their truth table with respect to their truth table there are a lot of things to be learned through the truth table so allow me now to go to the next slide where i'll be talking about the truth table of if p then q so in fact i will make the truth table for all of them but for that you have to understand one of them others can be easily be you know replicated over here so what i'll do i'll make a big table this is p this is q this is not p this is not q this is p implies q this is q implies p this is not p implies not q and finally whatever is left off okay so i'll write it down so this is for p this is for q this is for not p this is for not q this is p implies q this is q implies p this is not p implies not q q and this is not q implies not p. Now all of you please pay attention at least to the truth table of this. If you are able to understand the truth table of this you will be able to do all the remaining ones very very easily. So all of you please pay attention. Very very important. If p is true let me first put the truth values that p and q can take. So these are the four combinations. If p is true q is true then if p then q will be true.  So let's say you studied hard and you cleared JEE. Then will you come and say that sir whatever you said was wrong? Definitely not. You will say sir whatever you said was right. I studied hard and as per your saying I cleared JEE also. Thank you whatever you said was true. Correct? But let's say you studied hard you did not clear JEE. Okay? God forbid this should happen. So if you studied hard and you did not clear JEE the first thing you will come and say sir whatever you are saying was false. I put my heart and soul I studied almost 10 hours a day. I did all your DPPs. I did all the study material. I took the test honestly. I analyzed my mistakes. I did my self-assessment test but still I did not clear JEE. That means whatever you are saying was completely wrong sir. Isn't it? Yes. Isn't it? But let's say you did not study hard and you still cleared JEE. Does it make my statement any false or whatever I said was still true? What do you think? Should I write true here or should I write false here? Okay Rohan. Okay. Okay. So some are saying true. Some are saying false. The answer here is my statement still holds to be true because I never said that if you don't study you will qualify. So when your antecedent itself is not met you have no right to judge the consequence. Are you getting my point? So my statement was if you study hard but did you study hard? No. So don't judge my consequence then. So my statement is still a true statement. It is still valid. Getting my point. This is one of the surprising factors of the truth table of if P then K. Okay. If you did not study hard and you did not qualify JEE then is my statement true or is my statement false in that case? I would like to know your opinion. My statement is still true. Yes. Thank you everybody. Yes. It is still true because you again did not study hard. You can't come and argue with me. So my first thing is did you study hard? Correct? You said no. Then my statement is still true. No. Are you getting my point? So the only occasion when my statement will become false is when you studied hard and you still did not qualify JEE. That is the only time you will come and debate with me. You'll come and argue with me. Sir, whatever you said was wrong. But in the rest of the occasions you will not come to me or you'll come to me with a thank you. Are you getting my point? So this is very important. Now when people ask me sir how to remember this truth table I tell them a very simple analogy. I think I have told this to the naffle students also. So let's say I make a statement.  So let's say there's a husband who makes a promise to his wife that if I get a promotion, that is this is your P statement, then I will take you for a Europe trip. Okay. Let's say this is your consequent. Now try to analyze when is this husband a good husband? He gets a promotion, he takes his wife to Europe trip. Is he a good husband? If the answer is yes, write a true there. If the answer is no, write a false there. Okay. So he getting a promotion and he keeping up his promise of taking the wife for the Europe trip means he's a good husband. So true for that. If he doesn't get a, if he gets a promotion and he doesn't take his wife for the Europe trip, what do you think? Is he a good husband or a bad husband? Very bad husband, right? He did not fulfill his promise. So he is getting more money because he's promoted but still he is a conjuced fellow. He is not taking his wife for the Europe trip. Okay. So he's a bad husband, bad husband means false. Okay. Let's say if he doesn't get a promotion and still poor fellow takes his wife for the Europe trip. Okay. He'll say sir, he's a God husband, sir. He's a very, very good husband, right? Because even if he has no money, he's still taking his wife for the trip. So true there. Okay. Just an interesting way so that you remember this to table else, you will forget it. And let's say if he doesn't get a promotion and he doesn't take his wife, so please do not get surprised to see a true, true coming over here. I understand the first and the second situation, but many people forget the third and the fourth one. Okay. Now in the same hand, we can write down the truth table for others also. So first of all, let me fill up not P. Not P is false, false, true, true. Not Q will be false, true, false, true. Now to fill this up, let us fill this up now. Treat this as the antecedent, treat this as the consequent. Now did not get promotion, did not take his wife for the Singapore for Europe trip. Good husband. Did not get promotion, took his wife. Good husband. Got a promotion, did not take his wife. Bad husband. Got a promotion, took his wife, good husband, right? Similarly, for writing the truth table for this one, treat this as your consequent, treat this as your antecedent. Oh, sorry, what I'm saying. Treat this as your antecedent, treat this as your consequent. Okay. So now true, true, true, false, true, true. See, did not get a promotion, took his wife for Europe trip. True, false, false, false, false, true. Okay. Are you getting this point? Now let's fill up this one. Now treat this as your, treat this as your antecedent, treat this as your consequent. Didn't get promotion, didn't take his wife. Good husband. Got promotion, did not take his wife. Bad husband. Did not get promotion, still took his wife. Good husband. Got the promotion, took his wife. Good husband. Okay. Now why I made all these four truth table in one go is, what I wanted you to appreciate the fact is that the truth tables of these two are same. Do you see that? They have the same truth table. Correct? Yes or no? That means this statement is logically equivalent to saying this. A very, very important concept. A lot of questions have been, a lot of concepts are based around this. If P then Q is equivalent to saying if not Q, then not P. That means when I say, if you study hard, then you will qualify J is equivalent to saying you will not qualify J if you do not study hard. Got this point? So a statement or if then statement is logically equivalent to its contra positive. And at the same time you realize that the truth table for these two are same. Okay. That means Q implies P is logically equivalent to not P then not Q. Guys, essentially we are saying the same things. Here just P and Q positions are reversed. Okay. So the alphabets gets changed, but essentially I'm trying to say that a particular if then statement is always equivalent to its contra positive. Okay. Q to P again. Okay, we can see Q to P you treat this as your antecedent, the promise made by the husband and this as your consequent whether he takes his wife for the Euro trip or not. So see here, gets promotion, take his wife, good husband, doesn't get promotion, takes his wife again, a good husband, gets promotion, doesn't take his wife, then he's a bad husband, doesn't get a promotion, doesn't take his wife, still a good husband. Got it. Got it. Okay. Now, this is another platform where I want you to understand that the truth table of these two are not the same. That means if P then Q is not logically equivalent to if Q then P. Now this is very important because as youngsters, let's say ninth grader, especially me, I mean, I'll tell my story. I always used to frown at my teachers when they used to prove the converse separately. I'm sure most of you would be doing it. And most of you would be thinking that your teacher is just doing the same thing in reverse direction or something like that. But actually no, he was actually right in proving the converse separately because if this is true, it didn't mean this is true. Or if this is false, it didn't mean this is false. Okay. So these two are not logically equivalent. That is why in our junior classes, if you study, if you recall your circles, chapter, etc., where you had learned several circles properties, you would recall, you see that even if you visit those books right now, you'll be seeing that they will prove the converse also. Okay. So I used to feel very irritated why they are proving converse. It is the same thing. No, it is not the same thing. Had it be the same thing, their truth tables would have been the same over here. They are not the same by the way. Got the point? Any question? Any concern? So now before going for a break, quickly talking about the negation of negation of if p then q negation of if p then q is given by p and not q. That means these two are logically equivalent. Please note this down. So negation of if p then q is given by p and not q. Please before going for a break, verify this by truth table. All of you please verify this by truth table. I'm giving you around 90 seconds time. Not a big thing to do. So make a truth table for this guy. Make a truth table for this guy and check whether the truth tables are matching or not. If matching, tell me matching on the chat box. How are you finding this chapter? Easy. Okay, Archer. Thank you for confirming that. That was quick. Okay, Archer. Okay, Rohan. Good, Gayatri. Okay, let's quickly do it. I don't want to take too much time. I think I need p. I need q. I need one for negation q. I need one for... I'm drawing everything, but it is up to you. You can skip something because in je, they don't see your steps. So you can skip out some columns also, but do it at your own risk. And lastly, p and not q. Okay, so let's put the inputs through, through, through false, false, through false, false. Negation here will be false, through, false, through. Okay, now remember husband making promise to the wife. Gets promotion, takes his wife, good husband. Gets promotion, doesn't take his wife, bad husband. Doesn't get promotion, takes his wife, good husband. Doesn't get promotion, doesn't take his wife, good husband. Okay, so in some time, we will remember this truth table as well. Okay, negation of this will be false, true, false, false. Okay, now conjunction of this and this. True, false, false. True, true, true. False, false, false. False, true. False again. What do you see? Yes, exactly the same truth table. No difference. Okay, so they're same. So yes, it is logically equivalent. So I did two things over here. I first of all told you the negation also. And at the same time, I basically, I know tested you on your truth table making skills. Okay, so let's have a small break on the other side of the break. We'll talk about if and only if, and then we'll take up some questions from the previous year J exam. Okay. Okay, all of you enjoy your break. Okay, some of you are back. Good. All right. So in the interests of time, we'll continue with the last type of compound statement, which is called, which is called if and only if. Okay, if and only if statement are also called as biconditional implication. They're also called as biconditional implication. Okay. So when you say if and only if P, then Q, okay, I F F basically is a shortcut or you can say an abbreviated version of if and only if. So if and only if P, then Q basically means that both implies each other, right? It actually means both are logically equivalent to each other. So this will mean this and this will mean this. That means both are the antecedent and the consequence of one another. Right? So as an example, if I say if and only if I give, let's say, I take a name Mr. X banana, then he will eat. Okay. So what does this mean? It means that if you do not give banana to Mr. A or Mr. X, he will not eat or if he's eaten, you must have given Mr. A banana. Okay. So the fact that he has been served banana is equivalent to saying that he has eaten and the fact that if you do not give him banana, he will not eat. Are you getting my point? So both implies one another. So he eating or you giving him banana is equivalent to his eating. Okay. And he eating is equivalent to you giving him banana. Okay. So this kind of statement can be broken down as two if then statements connected by and operator or an and connector. So this and this basically will make your this statement. So you can say this is logically equivalent to saying this. No, if you give him a banana and if he doesn't eat, then these, you cannot say if and only if P then Q got it for a cool. So if you give Mr. X banana and if he refuses to eat, then this statement P and the statement Q are not equivalent to each other. That means you cannot say if and only if P then Q or in other words, it will become false. Got the point. So if he's given banana, he will definitely eat it. He cannot refuse to eat. Okay. So if you look at this definition, you can actually make the truth table for if and only if P then Q. So let's make a truth table. So, oh, sorry. Let me use my tools.  So P Q P implies Q Q implies P and the end of these two, the end of these two is basically what is your if and only if P then Q. So this is same as saying, this is same as saying if and only if P then Q. So let's see the truth table. True, true, true, false, false, true, false, false. So this will be true. This will be false. This will be true. This will be true. Here, treat Q as the antecedent P as the consequence. So true, true, false, true. Now you have to take the end of both of them. So true and true is true. False and true is false. True and false is false. True and true is true. Got this point. So here the conclusion is unless until both are true or unless until both are false, the statement will always be false. So only when both are true, it is true. Only when both are false, it is false. Rest if any one of them is true and other is false, if and only if P then Q will become false. That means if you give him banana and if he refuses to eat, then the statement will not be equivalent to this technique. This will be false statement. Okay.  Does it make sense to you? Does it make sense to you? Okay. Now, if I have to write the negation of such a statement, let's say I want to write down the negation of if and only if P then Q. It is as good as saying you are writing the negation of P implies Q and Q implies P. Correct? Now, remember when two statements are connected by an and connector, the negation of it will follow D Morgan's law. That means it will be negation of if P then Q or negation of if Q then P. And individually, we have seen the negation of this, which is P and not Q. Or this is Q and not P. Okay. So, this is your negation of if and only if P then Q. Now, do you have to remember this? If you want, you can remember. Else, you just have to remember the definition of if and only if P then Q and then use your normal negation rules of D Morgan's law and get the result. Okay. Please note it down. Now, the last part of this chapter is a theoretical concept which probably would be asked to you in schools. I would like to know from NPS R&R students, have you done the three types of validity tests for if then statement in your school? Direct method, contra positive method. Oh, this chapter is deleted only. Okay. So, what we'll do will not go into that because if that is not coming, then it is not going to help you in the J exam. Okay. So, we'll directly take some questions from the previous year papers. Just few of them, not more. This is J main 2019 question. It says that if you are born in India, then you are a citizen of India. What is the contra positive of this statement? I would like to know from you on this poll. One minute to answer this. This is super, super easy. Come on, guys. Nobody should leave it. Very good. Okay. Five, four, three, one. Okay. So, A and D have got most number of votes. But remember, one of you is wrong. One of you is right. See, if you are born in India, if P, then you are a citizen of India, then Q. Remember the contra positive of this statement, the contra positive of this is not Q implies not P. That means if you are not a citizen of India, then you were not born in India or you are not born in India, which is basically said by option number A. The last one, if you are not born in India, then you are not a citizen of India is actually the inverse of the statement. Okay. Get the difference. That's why I wrote them categorically converse inverse contra positive. Okay. So, remember, this is the contra positive. Again, I would write it down. This is contra positive. This is called converse. This is called inverse. Next, which of the following statements is not a tautology? Which of the following statements is not a tautology? Let's have two and a half minutes for this. Very good. Got a response. Okay. Five, four, three, two, one, go. Okay. Most of you have gone with C, option C. Okay. Let's check. Okay. So, guys, let's look at the first one. P and Q implies not P or Q. Okay. What we'll do is we'll make some two tables out of it. Let's say we make some two tables. P, Q. Let's see whatever we require. We require P, we require Q. We require P and Q I can see. We can require, we'll require, oh, sorry. We'll require P or Q also. And I think, oh, sorry. This was Q. And what else we require? We require not P, not Q also. We should have made that before. Okay. True, true, true, false, false, true, false, false. This is false, false, true, true, false, true, false, true. P and Q will be true, then false, false, false. This will be true, true, true, false. Okay. Now, not P or Q. Not P, not P or Q. So, this will be true, false, true, true. Now, P and Q implies this. So, true, true, true, false is true, false, true is true, true. So, this is a tautology. So, this cannot be your answer. Okay. So, at the same time, I'll keep deleting this because what I don't need, I'll just keep on erasing this. So, this is not your answer. Next is P and Q implies P. P and Q implies P. So, true, true is, I'll write it down. So, this is true, true, true, false, true, true, false, false, true, false, false again true. So, this is also a tautology. So, B cannot be your answer. Right? B is also a tautology. Okay. Next, implies P or Q. So, this is your antecedent. Here is this. Yeah. So, P implies P or Q. So, true, true, true, true, true again true. False, true, true, false, false is true. So, this also cannot be your answer because this is a tautology. Okay. Alright. So, let's erase this. Now, I think I need to make one more for the last one, which is P and, sorry, P or not Q. P or not Q. So, true or true is true, true, false, true. Now, we have to do P or Q, this, this, this, and this, we have to do, which is the last. Let me write it as D. So, true and true, true, true, true, true, false, false. Now, here itself it is false means it is not a tautology. Option number D is correct. So, as you can see, this question took us a little time. And hence, when you're doing such questions, please write minimalistic steps. Okay. Do not write all the things. It will lead to a big table and you'll get confused in that table. Is it fine? Any questions here? So, one last question we'll take and then we will move on to introduction to 3D geometry. Simple question. I think you should be able to answer it within 30 seconds. The negation of the statement, if I become a teacher, then I'll open a school. 5, 4, 3, 2, 1, go. And C only. C guys. Why C? I already told you, if P then Q, negation is P and not Q. That means I will become a teacher and I will not open a school. Option number A. It's already written in this language. If P then Q, negation is P and not Q. So, if I become a teacher and I will not open a school, why are you making mistakes? Nobody drops these kinds of questions here. They are very simple. Simple question. AI triple E, 2011 question. There are three statements here. So, if I become a teacher and I become a teacher and I become a teacher, nobody drops these kinds of questions here. They are very simple. Simple question. AI triple E, 2011 question. There are three statements here. Suman is brilliant. Suman is rich. Suman is honest. What is the negation of Suman is brilliant and dishonest if and only if Suman is rich? One minute for this. Yeah, it's very obvious. Good. Everybody should get the same answer. Okay, five, four, three, two, one. Go, go, go, go, go, go, go. Okay. Though many of you have answered A which is correct, but I'm disappointed with people answering C and D. So, Suman is brilliant. Suman is brilliant and dishonest if and only if Suman is rich. We have to write the negation of this. The moment you say this, option number A becomes correct. By the way, the positions of these two doesn't matter because both are antecedent and consequent of each other. So, even if you write it like this, hardly makes any difference to the statement. Okay. So, there's nothing like converse and inverse, etc. in a if and only if statement. Okay. So, you can see how easy questions are asked in this chapter and this is indeed a very scoring chapter and as all good things should come to an end, this chapter is also coming to an end. Okay. All right.


All right, you are still watching way. So National Pumpkin Day is celebrated on October 26th to celebrate or an appreciate pumpkins for their culinary and decorative uses. Pumpkins can be used to serve or in savory rather and sweet dishes and also popular autumn brevedges and Halloween decorations. Happy National Pumpkin Day. Why is it celebrated close to Halloween? I don't know, they used to do Halloween. But I want to say, have you tasted pumpkin soup? No. I haven't. Shout out like very sweet. I don't like sweet. No, it's not sweet. It's just something unique about that soup. Shout out to my blood, my sister, Ruth Lagarde. You know, she's French now. But what she's done with some of the dishes is she's infused a bit of the Nigerianness. I told her I'm going to open a restaurant and we sell out. Even they've bought. So there's this particular pumpkin soup she makes. Literally every day, I wake up some days, I'm just craving that soup. It's so nice. I don't know how to explain it, but that pumpkin soup is the best. So I think I like pumpkin. Because of that too, any other thing I've not tasted pumpkin. But that soup, I like pumpkin. I think for me it would have made sense if we had some pumpkin in the house or some of this pumpkin soup for us that we haven't. So we can taste that soup. We can't exactly tell. Yeah, sorry. But you're looking like pumpkin today because of the color of it. So we have a pumpkin live. Live pumpkin in the building. I don't think I've tasted it. But I'm sure it would be a delight because every time I see it like towards Halloween and all that, I'm always wondering what it tastes like because more often than not, you see it from a different, from a decorative point of view. So you're really, you're wondering, it's just like continuously seeing like a watermelon, doggy and everything. So you always wonder, you just wonder, okay, what it tastes like. It's not something that we have growing here. I'm sure it can be grown, but it's not... She used to make it when she was in Lagos, in Nigeria. I don't know how this works. Pumpkin is not... It's not even imported. It's not even imported. It's just like red oranges, like orange oranges. She goes to the supermarket. So she buys the imported ones. So I mean, but the soup is amazing. In fact, because of my sister's skills, Christmas and all these things like this, the indoors refuse to cook because you know the food they make is bland. You know, so it's just... Nice. No, not the truth. Even then they said they know. They don't abandon their food. They always wait for her to cook Christmas food because literally their food is bland. It's like kind of like, there's no... You know, Nigeria, ha! We have food. No, no, no. Our foods are like very flavorful, very, very rich. Seasoned. Very seasoned. Well seasoned. I mean, it's when you travel around, you realize that... Give it to us. Give it to us. You are missing a lot from home. Yeah. All right. So what do you find for us, Angel? So my new story is about the, you know, the alleged suspected yahoo boy who allegedly slaughtered his unipot girlfriend in Port Harcourt. So a man identified as Collins has been arrested for allegedly killing his girlfriend, identified as Justina along Road 15, NTA Road, Port Harcourt. According to reports, Collins was apprehended by his estate security personnel when he was trying to dispose her corpse after killing and extracting her eyes and breasts. Following the alarm, following the alarm raised by the security personnel, eyewitnesses stormed the suspect's apartment and found Justina's slaughtered body parts in sacks dripping with blood. Justina was a 300 level university, 300 level university student of Port Harcourt before her premature demise. So the state police command is yet to comment on the incident. You know, a couple of days ago, these things has been in the news for a while. A lot of killings, boyfriends killing their girlfriends, some suspected to be yahoo boys and for whatever reason, some other ones maybe ritual killing is always termed the same way. It's always targeted at the same thing. So at the end of the day, her eyes and her breasts were removed and obviously that's some level of ritual killing.  It's a 50-50.  You either get away with it or unhardly do you get away with it because somehow it'll catch up with you. So this is an unfortunate thing. I think a few days ago, there were videos of her last days with her boyfriend. I think there was a video that came out of two days before she was declared missing and only for her to be found. You can imagine how many days she... That's my thought process. You can imagine how many days she has been in his apartment for him to have performed all this because all this was not overnight. I think she has been missing for about three or four days. I think I heard it was her mom that raised the alarm. Yes, because she hadn't been seen after leaving the house. This is crazy. You know the other story that we had taken? They apprehended the guy in... Was it in Kenya now? I tried to find the place. They apprehended him and even came out on social media to say he killed his girlfriend. So he left evidence, yes. Very sad. Very sad. Please, I want to speak to young girls. Be careful who you date. Honestly, be careful. Because right now, when I see the way young girls go out, you're kind of reckless. You're really reckless with your life because sometimes we treat people the way we are, not the way they are. Because if you really look deep, you will see some of these traits in people. I'm being honest, right? So please be very careful who you hang out with. I mean, don't be by force to come out.  And the kind of person you date, some of these things. No, I'm telling you, these signs are there. They want to wear Gucci slippers. What's your problem? I'm working back. Well, they don't get to the working back. They end up in bags dripping with blood. They aspire. Yes, it's just very amazing what people do. Very sad. Sounds like your story. Oh, dear. So, Femi Aotearoa, business mogul, gifted 1 million each to 750 students of Augustine University on Thursday. That was when he was inaugurated as the chancellor of the institution. Where was that? You want to go back to school? I want to go back to Augustine University. I think I want to read the masters. In that school, right? But I think you already missed it. I mean, he was inaugurated. And that was today. So that was earlier today. And so 750 students each got 1 million. And so that's a total of 750 million. And then he went on to say, I read what he said, today I was officially inaugurated as the chancellor of Augustine University. I strongly believe in transforming establishment that I'm associated with. Therefore, to commemorate my appointment and in recognition of the harsh economic situation in that country, I gave a gift of 1 million to each of the 750 students. I hope this donation of 750 million NARA assists the plight of the parents of our students in this difficult time. Signed F. Watedola. F. Oh! This is so... Did you say he was the vice chancellor? Yes, vice chancellor, yeah. Well, that's interesting. Congratulations to... Yeah, that's quite interesting. VC of the university. That's interesting. All right, so the House of Rep members on Wednesday to yesterday that they rejected emotion, urging members to adopt at least one public school in their localities for the purpose of paying registration fees for the students sitting the unified tertiary multiplication exam. The WIEC, that's the West African Senior Certificate Examination and of course the National Examination Council. That's NECO, WIEC and JAMP and UTME. So trouble started when a member representing acerco-federal constituents in a dual state, Annamero, DeCary, moved the motion on the need to compel JAMP, WIEC and NECO examination bodies to register students for free in 2023 to 2024 examination exercise. He premised his motion on the need to assist poor parents in the face of harsh economic realities in the country. So on one hand, FEMI authorised gifting, a million Naira to 750 people. On the other hand our reps are fighting against paying, was it called giving, JAMP, WIEC and UTME for free. Interesting. Now I'll ask my sister, how much is the registration fee? How much is it but we're buying cards? That's why I took the story now. That's why I took the story. So we'll take a break. Let's discuss the SUV. Since people don't know what to do, WIEC for free. At least let's discuss the SUV. Stay with us, we'll be right back.


You've come to the right place in this video We have sourced the market and we have found the best 25 products that are sure to dazzle your customers and boost your sales So join me and let's begin with product number one to kick off our list We have this man's wolf necklace It symbolizes strength and courage and it's a very meaningful accessory to complete a stylish statement It's safe to say this item is a staple to have in your dropshipping jewelry store So let's talk a bit about the profitability of this product So as you can see here on the aliexpress page This product can be sourced between five and six dollars shipping included to United States And as you can see from this website that I found right here, you can easily sell this for $30 This gives us a potential profit of around $22 which is amazing for such a product for ads You could use something like this and of course image ads work Amazingly well if you know how to use them This is a good example of how to use image ads you split all of the real estate in as many pictures as you can to show Your buyer multiple angles of the same product This is a very good alternative to a video ad and as you can see from their ad they have more than 1.3,000 reactions with these figures we're pretty confident that they are doing amazing So if you want to sell this product all you have to do is go to aliexpress get the pictures from there Make a nice description with the help of chat GPT and you're all set next on our list are these magnetic therapy Earrings these precious earrings are magnetic So no one would ever know that they are not real plus they have special health benefits with this set This product would surely make an impact for the people that don't want to have their ears pierced because it's painful I'm one of them. Let's talk a bit about this product's profit potential as you can see from here The product costs around three dollars to source and according to this seller You can easily sell this for twenty five dollars the potential profit for this product stays amazing twenty two dollars Let's take a look at the competitors and as you can see they are going the review path So they are basically showcasing a review from an actual customer while the pinned comment right here shows a 90-day return policy and discounts up to 80% I would highly encourage you to use this in the video captions, but that doesn't matter the ad has 1.1 million views guys 1.1 million views close to 5,000 reactions and 1.6k comments Just imagine this product has widespread popularity now the website that we found right here shows the earrings with free shipping Which is a really good thing for such a product You want to give small products with free shipping because you can afford to do so Usually the shipping for such a small product is so small that it does not even matter So it would be nice for you to give it away for free because the client would highly appreciate that another good thing That these guys are doing is that they are offering bundles thus increasing the AOV AOV stands for average order value So if you can make your clients buy more than one product you will increase that AOV now I'm not a big fan of the fact that they are using so many bundles because you can get lost in those you want to close Your sales as fast as possible and you don't want to confuse your visitors more over the custom rating for this product is five Star meaning that the customer trust this and the future customers will also trust this on the suppliers page on Aliexpress You can see that this product is completely sold out meaning that people are going crazy over this product Although we might not be able to source this product from Aliexpress There are plenty of high quality pictures in there so you can dig in there and use them for your own store Another magnetic product from our list is product number three this heart shaped pendant necklace Not only this sparkly accessory is beautiful as a string of hearts It also converts into a four leaf clover. It's a very unique product That's catching the attention of a lot of buyers out there Now let's talk a bit about profits for this product as you can see right here You can get it for around three dollars shipping included to united states And you can easily sell this for around twenty three dollars Which leaves us to a profit margin of around twenty dollars, which is great I would usually target female audiences between the age of 20 and 50 which are married and unmarried But also men because they usually give presents to their girlfriends or their wives in this ad that I found right here The seller cleverly uses the functionality of the product in a video ad and the caption also includes a shopping link For the buyer's convenience more over this ad has an excellent engagement ad if you look at it They have more than 27 000 reactions and more than three million views The seller that sells this product sells it for 2290 with shipping included And it's also available with a gift box option, which is a good thing to do with such a product Also, they are using a really good scarcity technique Which tells you that it's the last day from which you can get a 49 discount Of course, it's not gonna be the last day if you come back tomorrow It's gonna be the same but it is a really good incentive for the visitor of your website To make them think twice before leaving your store without buying on the supplier's website right here As you can see the product is available for around three and a half dollars And you have four variations of the same product which will give your customers more options to choose from Plus on this aliexpress sellers product page You can find a lot of information about the product that you could definitely use for your store You can just copy and paste it to your store or edit it to your liking Also, they are offering a 14 day delivery for products that are more than five dollars Which is pretty good and a 75 day buyer protection And you can take advantage of that to offer your customer's protection Next up we have product number four This romantic necklace with gift set is an amazing addition to any dropshipping store that's selling fine jewelry This item is on the rise and people are choosing it because it's basically the perfect option for valentine's day Christmas mother's day so on and so forth So let's examine the benefits of selling such a product This ad right here showcases the beauty of such a product and it amassed more than 3.5 million views guys That's already a massive audience of potentially interested people to buy this product Plus the engagement rate is also amazing with more than 11 000 likes However, they only highlighted the link in which you can buy this product and a pinned comment Which is not a good strategy You want to include that link in the description of the product in the caption of the product Because you want to lock your clients. You don't want them to get lost You don't want them to not know where to buy the product from and if you're just doing this in the comments people might not find it So these guys are definitely leaving some money on the table on the seller's website This product sells for 38 dollars including shipping We can already see how it's available in three different styling options two gift box options and two color options They also do worldwide shipping fast return and 90 day money bag guarantee in essence Transparent business policies will always increase the conversion rate of your store. So make sure to do that on the aliexpress website This product sells for around 16 dollars But it gives us limited options to choose from nevertheless There are a lot of product information available on the aliexpress website and you can use those as guidance for your store Also, you can use their pictures next product on our list is this sunflower fidget ring Which is a beautiful and oppolite rendition to the almighty fidget toy It's not only an attractive piece of jewelry, but it's also a useful fidget distraction for those who need it So this product uses both style and function. Let's dig in the potential profit of this product Now first let's start with the ad this facebook ad that we found for this product markets the product as a symbol For joy beauty and warmth Notably this video has more than 1.3 million views and hundreds over hundreds of reactions With such high engagement, we can almost guarantee that this product is doing pretty well on this website that we found selling this product They are selling it for 31 dollars and the product description as you can see tries to create a personal connection between the customer and the website This store has also well coordinated colors giving us a really nice feeling when navigating it Additionally, as you can see here with the color orange on the header announcement They are offering free shipping for orders more than 35 dollars Which is a pretty clever marketing strategy because this product is only 26 dollars So it kind of forces you to buy more than one product if you want to get advantage of the free shipping On the suppliers website on the other hand side on aliexpress They are offering this product for 3 dollars and 78 cents with free shipping plus they have 13 different variations Now if you didn't notice the website that we found only uses one single variation of this product Which is a really good thing to do if you want to keep your customers focused on one single variant on aliexpress You have 13 different variants to choose from so you can put this in balance product number six this wavy apple watch strap This product is a subtle approach to jewelry which attracts a more matured audience It's a very hot selling product which allows the user to customize their apple watch by using this product You can have a more subtle sophisticated look while wearing your apple watch It's definitely a great product to add on your store and store selling now Let's talk a bit about money this ad that we found right here highlights Its impressive features and attractive design. They also promote a buy one get one free offer Moreover as opposed to some other sellers this seller actually provides a buying link So you can just click that link and buy the product making the process a lot easier for potential buyers The video itself has more than 2.2 thousand likes and 182 shares which makes it pretty popular on the sellers website This product cost 69 90 with free shipping the product is available in five different color variations And it is showcased through high quality images Plus as I told you they are currently offering a buy one get one free offer Which will surely make people buy in addition They are including customer feedback which will surely incentivize people to buy It will give them more courage to buy knowing that somebody has bought the product before them from the same website On aliexpress, they're selling the product for 13 25 with free shipping And they're guaranteeing that you will get the product in 14 days at this price point We can safely say that you can profit up to like 56 dollars per product Moreover, you can find a lot of high quality images on aliexpress But what's but the really convenient thing you can find it in the bottom of the page in the customer reviews that you can import for your own store Another trending drop shipping jewelry product that you can sell right now is this trending abstract hug necklace This is a popular gift to express support and attention Moreover, this space right here deserves a special spot on our list because it is pretty unique And it can hold a lot of sentimental value with that said, let's take a look at the profit potential Now first things first this ad that we found is a single image ad They are only using one single image of the ad promoting this product This is a good idea if you don't have any alternatives You can just not complicate anything and use a high quality picture one single picture for your ad They're using the beautiful text. There's no one like you underneath the photo of the product As you can see, they have a buying link in the description of the product Which is really good and this ad has more than 8.5 reaction Which means that it's a pretty popular product on their store This necklace sells for close to 40 dollars Besides for the highly trending black and silver product This seller is offering multiple variants when drop shipping jewelry products We must consider the materials that they are made from that's why the seller of this product States the fact that this product is free from nickel lead and caldmium This means that this product is generally safe to use for all kinds of different users And they are also offering free worldwide shipping Which is really good thing to do for such a small product checking out aliexpress We can source this product for two dollars and 39 cents shipping included on the downside of this product They're offering multiple languages to choose from so you gotta be really careful about that Nonetheless, this product gets like a 4.8 out of 5 portraying its top caliber quality Joining our trending drop shipping jewelry finds is this glowing moonstone necklace In asian times this moonstone necklace was believed to be connected with the moon You've guessed it right. This means that it brings protection and luck to its owner That might be right But today this moonstone necklace is a mystical accessory that can attract significant traffic on our drop shipping store Now this facebook ad that we found right here enchants us with its magical description The ad image highlights the mystical glow of this product Giving us an edge and giving us an advantage because the client will see this product being in use Will see this product is glowing in the dark. So basically will help all customers see how does this product work in real life In fact, many customers do believe that this product is actually magic This is why it has more than 100 000 reactions. That's absolutely crazy Now if we go on their website, you can see that they are selling this product for 40 dollars actually 39 99 They're also committed to environmental sustainability and they are promising to plant a tree for each and every product So that's pretty cool, right? Now some good news. You don't have to travel far to find this product You can find it on aliexpress as you can see right here for only two dollars and 74 cents That's a steal for this product given the fact that this website sells it for 40 dollars It gives them a profit margin of more than 37 dollars However on aliexpress the product description and images are not that good So you can do a much better job by for example going to chat gpt and writing a good product description Just be careful. You need to create a very nice product description That is what highlights these products because they are pretty blend usually So you need to highlight them with the product description and chat gpt does a very nice job with that At number nine, we have this natural stone crystal pendant right here the mineral that it's made of as you can see right here It's supposed to have a healing power and an energy amplifier inside of them So they say now besides that this looks like a really cool product and it looks good on basically any outfit This is why this product is such a good choice I have a really good tip for you include various types of materials and crystals for this specific product Because each and every crystal holds a different significance for its wearer And you remember I told you earlier in this video to not use many variants of the same product and many colors because people Will get confused This does not apply for such a product because each and every crystal is different for the person that buys it 10 we have yes, you're right hand of fahtima pendant This piece of jewelry has intricate patterns symbolizing meaningful ideas This is why it's such a good fit for many different cultures Furthermore, this product can be made from stainless steel brass silver or gold and number 11 We have this heart pendant necklace right here now This my friends is one of the most sought after products in this drop shipping accessory realm And it's actually one of the most in-demand jewelry products of all time Google trends shows us a constant popularity of this item this heart pendant has a lot of symbols But mostly they levitate around love. So who doesn't love you can sell this to anybody You can sell this to mother's day sister's day birthdays presents Random presents valentine's day christmas, etc, etc This item includes a variety of styles and undeniably this product will attract a lot of eyes from shapes to characters This letter pendant chain right here is a guaranteed eye turning product with a lot of style and versatility This item can be made from wood crystal silver gold and stainless steel If you seek a one-of-a-kind gift for your loved ones, this product is the right choice Just a very quick tip you can personalize this product with various options and you can engrave stones in them or different patterns Are you looking for a timeless piece of jewelry to sell on your store? If so, then this product right here is the right choice for you This shiny silver choker right here will add a touch of elegance to any outfit you can imagine This item's popularity on google trends is evident. We cannot contest that so by dropshipping this product We can generate substantial profits with its mesmerizing colors and effects This opal star necklace is one of the best-selling dropshipping fashion accessories products out there Opal gemstones and star shapes are popular representations of hope and creativity That is why this product is such a good choice for those who love celestial themes now for number 15 Let's check out this wood beater choker ranging from 14 to 16 inches of length This dropshipping piece of jewelry can be painted or can be customized any way you want When selling this item, we can just target bohemian fashionistas or just people who love natural stuff for number 16 This item is basically sold to anyone any age any gender any sex Any whatever this white beaded bracelet as a popular dropshipping accessory for all genders and ages This white beaded bracelet is basically a must have for any dropshipping store out there It's a perfect accessory for any outdoor event Or why not if you're going to the beach you can also wear this The fact that this product is neutral yet versatile at the same time Selling this product can result in a high sales potential This thin gold enclave right here is a famous social symbol for wealth and status We have all seen this product in the past I'm sure you have if you didn't know This product is also a meaningful accessory to religions all around the world And today the popularity for this product is constantly growing So you gotta check it out joining our list of eccentric dropshipping jewelry pieces Is this evil eye bracelet right here and this product features a talisman in the form of an eye This eye is supposed to save you from the evil eye Today is a symbol of luck and safety and it's a very popular item all around the world next up number 19 We have this rotating beads ring. So we had the rotating beads bracelet now We need to talk a bit about the ring interestingly enough the beads that this ring is made from Can be made from different kinds of materials such as metal plastic wood or acrylic aside from being just a fashion tool This product is also a calming tool at number 20 We have mold of it which is a pretty rare and unique gemstone And this mold of its stone ring right here is a pretty sought after product Besides the fashion purpose that it holds It also brings tranquility to its user and it's said to bring positive changes in our lives And my recommendation for you if you want to sell this product You can sell it for one single stone or multiple stones arranged the way you want to Another classy add-on to our dropshipping jewelry business is this product right here this black tassel earring This accessory comes in different tassel lengths thickness and embellishments Remember to dropship only long-lasting and high quality tassels for the comfort of your buyer's years the popularity of these hoop earrings Is continuously rising and because of its versatility is one of the most sought after products for women It is suitable for casual and formal occasions as well And this product is a top dropshipping fashion accessory to sell right now Additionally, this item can be made for various materials such as plastic gemstones or silver for a more stylish look This item can be incorporated with other products as well And those products are beads crystals or engravings at number 23 We have this heart statement earrings which are a must for customers with bold fashion tastes This everyday wear accessory will come in different shapes and sizes lock types and styles This is why it's safe to say that this piece of jewelry is such an in-demand product right now According to google trends right now This product is at an all-time high and i'm talking about worldwide This is why selling this product right now can be such a good idea Now this piece of jewelry is more directed to people who take a more subtle approach to fashion And you can bring this in endless styles Our last dropshipping jewelry item from this list is this small hoop earring set Interestingly enough this piece of jewelry contains Embellished little hoops inside of it. So before sourcing this item Just make sure that you're sourcing hypoallergenic materials to ensure your customer safety Now that we have 25 excellent products to sell for our dropshipping jewelry store The next step is to choose a reliable source for them One tip that i can give you here is the fact that if you're integrating with multiple suppliers You're protecting yourself against one supplier not having the product and this way you don't have to rely on a single source So if one supplier runs out of stock or i don't know what happens. They raise the prices We are covered another advantage by working with multiple supplier is diversification More sources mean a wider variety of products This will allow us to stand out from the competition Moreover, this approach is an excellent way to gain more dropshipping experience because each supplier has unique policies Products prices shipping methods shipping times so on and so forth So here are some key suppliers that i want to talk about etsy amazon aliexpress Walmart banggood and much more with auto DS is more than 25 dropshipping suppliers We'll have a lot of opportunities to access millions of products worldwide and manage our business effectively Furthermore, the auto DS is all in one dropshipping solution will help us diversify our product listings and expand our market research


Dear students, I would like to present to you an example of computation of probabilities that cannot be found through the marginal PDFs. So let us look at this interesting situation. Suppose that the joint PDF of two continuous random variables x1 and x2 is given by small f of x1, x2 is equal to 4 x1 into x2, where x1 is going from 0 to 1 and x2 is also going from 0 to 1. Now if we want to take out this probability that x1 lies between 0 and 1 by 2 and x2 lies between 1 by 4 and 1, it is quite simple. It is quite simple. You will simply put the limits of the double integral according to what you have and you will solve it step by step and you will get it. For example, if we write it as integral 1 by 4 to 1 and then the next integral 0 to half and then f of x1, x2, 4 x1, x2. After that, you have to write very carefully that which of the dx1 and dx2 you have to write first. Since you had written 1 by 4 to 1 in the first integral, those are the limits of your x2. Therefore, you will write the most at the end, dx2. And what you had written in the second, i.e. 0 to half, are the limits of your x1. Therefore, you will write dx2 at the end, but before that you will write dx1. Because you know from calculus that if there is a double integral, then first you solve the integral inside and then you solve the integral outside.  And of course, you can do them yourself and doing them. What is the final answer? It is 15 by 64 and that is the probability of this particular event. Suppose instead that we are wanting to find the probability that x1 is equal to x2. Well, the probability of x1 is equal to x2 can be rewritten as the probability that x1 minus x2 is equal to 0. Now, since the random variables x1 and x2 are continuous random variables, therefore their difference is also, my dear students, a continuous random variable. This difference, this is also a continuous random variable. But the point that you have to understand here is that if we are saying that we are wanting to compute the probability that this new random variable x1 minus x2, whether you name it y or w or any other name, that this new random variable is equal to 0, so the thing to think about is that 0 is a constant. And we know by the basic definitions and all those things which are attached with it that the probability of a continuous random variable being equal to a constant is always 0. Continuous random variable, it is not discrete. Continuous random variable, it is always in an interval. So therefore, the probability of x1 minus x2 is equal to 0. In other words, the probability of x1 is equal to x2. This probability is equal to 0. This was another interesting point and now, importantly, let's take out another probability. And you will see that here it is slightly different and it is not like the very simple case when we take out a very simple distribution. So this is this one. When we are wanting to find the probability that x1 is less than x2. If we want to take this out, then how do we proceed? Students, in this case, we will write it as follows. The first integral from 0 to 1 or end we are writing dx2. That is, the entire range of the x2 variable in other words, according to our domain x2, we are taking 0 to 1. However, for the second integral, we are writing 0 to x2 and you must realize that it is alright because we also had to take out the probability of x1 is less than x2. So now this second integral is according to x1 because it was the first one according to x2. So now if we are talking about x1, then we want it to be less than x2. Which means that the integral has to go from 0 from the starting point up to x2 only. And then after that, we write down inside 4x1x2 which is the expression of our joint PDF. Alright, now that we've got all this, now we can simply solve it. First you solve the inside one and keep in mind that if we are solving it according to x1, so x2 is acting as a constant and then you do all the steps as you can also see on the screen. And once you have got that, after that you solve the outside integral which is according to x2. And doing all those steps, you obtain this particular probability equal to 1 by 2. So this is the procedure by which you are able to solve sometimes apparently somewhat complicated situations also. You just have to be a little careful as to how you put the limits to your integrals and also then solve accordingly.


It's been a wonderful afternoon and we're going to get on with that for on joints in your direction. Here's David who just signed. So I'm going to teach you how to hand out what we got 10 pounds of gear, okay. The ground has been dry, there's only been a little bit of drizzle all day, so the ground is dry. We need a pathway that goes right from the middle all the way to the first tent there and then to pass that way.  So I need some plot head, take two steps back that way, take everybody take two steps back that way and we need a path end all the way to the end.


Dating in today's world is so complicated. There are so many rules, regulations, and hidden meanings that it's almost necessary to get a manual when you first become attracted to someone. It's scary enough trying to find someone that wants the same thing as you, whatever that may be, without worrying about the million other things that could happen to make such a fantastic experience go wrong. For some reason, today's society views suddenly disappearing from someone's life as acceptable behavior in the dating realm. As many of you know and unfortunately experienced, this concept has been dubbed ghosting, or in action form, to ghost. This concept more or less applies to those who have been involved with the significant other for some amount of time, from seeing each other for a few weeks to being together for months to even longer. This can also happen between close friends, but is especially prominent in the dating world. This awful practice appears to be happening more and more often, but as anyone who knows the basics of psychology can tell you, just because you run away from your problems does not mean they go away. Obviously, having that special someone suddenly beginning to ignore your messages and calls can be devastating. You were so into them, and you thought they were into you, but then they do this? What's up with that twisted little bit of guiltlessness? As someone who has been ghosted before can verify, you feel like a wrecking ball has blasted your self-confidence and self-esteem, especially if the person who ghosted you was your official significant other. Breaking up with someone is hard enough, but this adds even more insult to injury. Quite literally, in fact, this practice is insulting to the receiver. It's as if you weren't even worth being properly let go. It fits in nicely with other forms of social rejection, which lights up the same areas of the brain as experiencing physical pain would. Why do people think it's okay to ghost? Why would someone who claims to care for you suddenly send you head first into Singletown without explanation, or even grace you with a five-minute notice? Many think it's a result of fearing conflict when having to confront other people. Ghosting is a convenient way to completely avoid this. Why go through breaking up with someone when you can just bypass that? Researchers have found that using this method of avoidance to break up with someone is one of the worst ways you could end a relationship. The person being ghosted feels most of the blow, but it can also affect the person who did the ghosting. Many of us don't take that kind of crap from people, and in a fit of rage confront the person. This can be especially distressing or humiliating for the ghoster if the ghosty knows where they exercise, live, work, etc. Not to say that they don't deserve this, but it does cause problems and emotional stress for both parties nonetheless. Many feel that the person who did the ghosting probably doesn't care about them anymore since it was so easy for them to disappear abruptly. Numerous professionals disagree, saying that the guilt from taking such a spineless way out of a relationship can have negative long-lasting effects on the person who did it. This behavior also further strengthens the distress or anxiety that confrontation may cause. As mentioned before, running away from someone doesn't necessarily cause it to stop chasing you. Breaking up with someone properly through open communication, no matter how hard it may seem, is the best thing to do, even though it may feel like you're ruining someone's life. It's best to confront your problems head on and try to resolve and overcome them. Have you ever been ghosted or ghosted someone? How did it make you feel? Do you think the ghoster did so because they are weak or because they are heartless? Share your comments below. Thank you for watching, and don't forget to subscribe.