Welcome back in the last few lectures we talked about many important decision procedure methods starting with the truth table method and then semantic tableaux method and then we have come up with semantic resolution refutation method etc and all when we solve one particular problem with respect to all these methods and then we have seen which method to we need to employ and all it is all depends upon our convenience you know. So far we talked about the different decision procedure methods with which you will come to know whether a given well-formed formula is a valid formula or tautology or when two groups of statements are consistent to each other etc. So today I will be talking about another way of doing it which comes under the category of syntactic method so that is the axiomatic prepositional logic I will be talking about the axiomatic prepositional logic in that I will be talking about two different axiomatic systems one is due to the famous Russell and Whitehead there are two mathematicians and all this axiomatic system is due to Russell and Whitehead which is called as Russell and Whitehead axiomatic system which they proposed it in the historic book that is principia matematica. So I will be taking a selective portion of the principia matematica and I will be talking some of the proofs which are already there in that book so this comes under one particular chapter which is based on deduction in prepositional logic so I am restricting my attention to my focus on prepositional logic. So we will be mostly talking about Russell and Whitehead axiomatic system so now in this lecture what I will be doing is these are the things which I would be doing so first I will talk about what I mean by an axiomatic system how it has originated etc and then within the axiomatic system what occupies the central position is a proof of what a given theorem so we will be talking about these specific key terms such as axiom prove what you mean by deduction what you mean by saying that a particular preposition particular statement is considered to be a lemma or conjecture or corollary etc these are the things which you commonly come across within the proof theory. So usually a proof is considered to be a finite sequence of steps which ends in finite intervals of time so if your proof ends in finite steps in finite intervals of time then usually it is considered as an effective proof so how did we come to this particular kind of rigorous proofs and all so what is wrong with the proofs that are already existing in the Euclidean geometry Euclidean geometry is also considered to be one of the important axiomatic systems but still we do not treat it as a as rigorous as this that we find it in in the principia mathematical so what is the reason for that particular kind of thing I will talk about it in a in a nutshell and then I move on to a different kind of axiomatic system which derives its motivation from axiomatizing geometry principia mathematical is motivated by motivation has come from the arithmetic whereas Hilbert Ackerman axiomatic system the motivation comes from the geometry so then we will see with the help of deduction theorem will be simplifying some of the proofs that are there in either in Russell weighted axiomatic system or in the Hilbert Ackerman axiomatic system and then we will talk about some of the important meta theoretic theorems such as soundness so that is whether all the things that you are proving are going to be true or not usually in the proof the last step is considered to be a proof which is obvious obviously considered to be true if it is a true or tautology then it has to find a some kind of decision procedure we need to find a distant with the help of decision procedure method we should be in a position to check whether it is a valid formula or not the soundness ensures that all the things that you have derived are true and the completeness assures us that all the true propositions finds some kind of proof and consistency is a natural property which is in any given point of time you cannot derive both x and nautics so in this lecture I will be focusing my attention on the origin of axiomatic system and then partly I will be talking about Russell weighted axiomatic system and some of the important proofs within that axiomatic system due to Russell and Whitehead so the origin of axiomatic system is usually like this usually we find axiomatic system either in Aristotelian framework or even in the Euclid's in the Euclid in the works of Euclid that is elements so Euclid has come up with five postulates and then the few definitions and then there are some common notions and with the help of these common notions definitions and axioms he consider he use it as postulates with that all the other true propositions are true propositions in geometry are derived. So in the 90 in the mid 19th century the mathematicians started finding flaws in the Euclidian axioms so there is fifth postulates which is which is objectionable so the flaws in the Euclidian axioms in the geometry led to this particular kind of axiomatic systems so the work of cards in 1998 he found flaw in the Euclidian axioms is the postulates usually the fifth postulates is highly debated in on whether or not to replace that particular kind of postulate or change it a change it in a such a way that you can talk about a different kind of geometry. So after in the mid 19th century different geometries have come into existence Riemann geometry Cauchy's Gauss has come up with another kind of thing etc all this things comes under the category of non Euclidian geometry. So in addition to the independence of the parallel postulates established by another Russian mathematician Lopschewski mathematician discovered that certain theorems taken the certain theorems in Euclidian geometry are taken for granted by Euclid were not in fact provable by using his own axioms so there are two problems with this Euclidian geometry the first thing is that the fifth postulates postulate has created problem and people have come up with various other kinds of geometry and all and the other thing is that there are so many common sensical notions direct observations etc which are not part of part and parcel of your proof they are already present in the proof and there are certain things which are outside the proof and unfortunately they are also part of the proof and so we need to separate all this common if you want to have a rigorous kind of proof everything needs to be stated explicitly and there should not be any notion which is not part of the proof that should play a role in deriving some kind of theorems. So among this is a theorem that this is the fifth postulates which tells us that a line contains at least two points are that circles of some same radius whose sentence are so whose centers are separated by that radius must intersect so this has created a problem in all and then it led to different kinds of theories whether the angles total sides of the angles of a triangle angle of a triangle is considered to be 180 or greater than 180 all these questions arose so this is one thing which led to think in a different way the non-euclidean geometry or making the proofs more rigorous and all that was Hilbert was taking up in his axiomatic system and the other thing is that there are concerns that mathematics had been built on not been built on proper foundations. So Fregge has built it built an axiomatic system so that led to some kind of paradoxes we usually mathematics was rest in on set the but set theory is plagued by important paradoxes which was discovered by Russell invited and this paradoxes plague this kind of axiomatic system that also led to the development of rigorous axiomatic system that is what you find it in principia mathematical there are better Russell and Whitehead's motivation has come from some of the paradoxes that arose in the work of Fregge a set theory is plagued with paradoxes right and there was issues in the geometry and then Hilbert Ackerman took geometry into consideration and axiomatize geometry and all these discovery of paradoxes in informal set theory cause someone to wonder whether mathematics itself is consistent or not and to look for proofs of consistency etc and all. So there are two things which we need to note the developments of Euclidean geometry led to some kind of rigorous kind of axiomatic system which is little bit different from what Euclidean has done that is in your proof if you proof has to be rigorous all the common sensical notions direct observations etc should not be it might be mistaken all we might it might be very well be the case that our intuitions might be misleading you know so in that sense everything has to be stated explicitly and then from that all the truth things can be derived in all so for that you need to come up with a few axioms as much as possible and then few transformation rules etc and then you derive all the other true things so that is what we have discussed so far we discussed syntactic and semantic decision procedures semantic decision presence of truth table semantic tableau etc and all syntactic procedures we have natural deduction etc and all and another method added to that we have this axiomatic propositional logic where we are not interested in talking about meaning of this formulas and all but we will be interested in some of the patterns etc and all so given the axioms we will be deriving some kind of theorems so now in this this is axiomatic propositional calculus comes under the category of syntactic kind of proofs and all which we will be talking about in greater detail so in this we intend that proofs in propositional logic should provide demonstrations of formal truths that means you are deriving all the true propositions from a given set of you can call it as self evident truths or things which cannot be proved etc and all they are all axioms and derivation should provide deductions of the formal sequences of assumptions if that is the case then that is what we are trying to do that means in a nutshell all the true propositions so that you come across all the valid formulas now we are trying to generate proofs for those things it is not that they came just like that but as an outcome of some of the axioms and the transformation rules you will generate these theorems in addition we would like our formal system to be rich enough for all formally true formulas of L to be provable that means all the two propositions that you come across in your formal axiomatic system have to be provable it has to find a proof you say that something is true and you do not have a proof that would not work in all so you need to have a kind of proof for all the two propositions that ensures that all the two propositions will find a proof and for a well-formed formula A to be derived from a set of formulas gamma whenever the argument from gamma to A is usually considered as formally valid all the two propositions also have to be valid so now these are some of the examples of axiomatic systems principia magnetic and Hilbert Ackermann axiomatic system there may be many other axiomatic systems but I will be talking about mainly these two axiomatic systems so to begin with what do you mean by an axiomatic system so an axiomatic system as you can take it as a Hilbert Ackermann axiomatic system or principia magnetic that is Russell whitehead axiomatic system it is defined as follows at least these are the four five things which essentially any axiomatic system should consist of the first thing is that a set of well-formed formulas are usually called as axioms these axioms need not have to be proved and all so there are considered to be self-evident truths which are obviously true always true and all and then these axioms will also serve as schemas schemas in a sense that you substitute anything into it uniformly you will generate a theorem so that is the greatness of this axioms and these things does not require any proof that is why they are well self-evident kind of truths in addition to that there are some set of rules of inference which licenses some kind of operations on well-formed formulas of propositional calculus that means whatever you intend to have as an operation will not work in all but there are some set of rules like rules of inference such as modus ponens transformation rules such as if you uniformly substitute some formula into the formula then also you will generate theorems from the axioms so this is the second thing you need to have transformation rules such as modus ponens etc and all as minimal as possible in any given axiomatic system these rules would be very minimal and the third thing you need to have a set of well-formed formulas in the propositional logic propositional calculus which can be obtained from the axioms by means of rules of inference that is usually only one rule of inference is usually used that is the modus ponens rule so when you have a in plus b and a gets detached and whatever you whatever follows is b that is the main rule of inference which we commonly use in all the in throughout the axiomatic systems whatever axiomatic system you are trying to consider the fourth one these new well-formed formulas are usually called as theorems of the system that means you started with the axioms and then you transform those axioms are you trim those axioms in such a way that by applying transformation rules and truth-preserving rules such as modus ponens you generated another kind of inference so our path is like this that our path is like each and every step starts with the truth that is the axiom which is obviously true and then you if you apply transformation rules on these axioms and the resultant well-formed formula is also going to be true and then that gets transformed into another well-formed formula that is also going to be true each step is considered to be true so in that sense the last step of your proof is also going to be true if all the steps that are there in the deduction process is already true and the last step of your proof which is usually called as a theorem is obviously considered to be true so establishing the validity is another kind of issue but deriving kind of true formula from a given sequence of formulas is another issue so lastly that we use a particular kind of term that is what we call it as thesis thesis is like some kind of it can be taken as either axiom or it can be even taken as a theorem so in order to say that any well-formed formula is axiom or even a theorem sometimes we use this word thesis so every thesis is obviously considered to be a valid well-formed formula of a given field and every valid formula of that field is also considered to be a thesis of the system either it should be an axiom or it should be a theorem all the valid formulas it should be one of the one of the things should be true either it should be an axiom which doesn't require any proof or it can be proved by reducing the given axioms by using transformation rules and rules of inference to another kind of theorem and all. So this is what we mean by an axiomatic system so in a nutshell an axiomatic system should consist of at least set of axioms to start with either you can have three or you can have four or you can have five but as minimal as possible to start with and then you apply transformation rules substitution rules etc and all you transform these axioms into another thing which are also considered to be theorems and then you need to have some kind of rule of inference that is usually we use it modus ponens that's all you want to derive all the two propositions that occurs in your formal logical system. So now there are certain terms which needs to be defined clearly before moving further otherwise it will create some kind of confusion so the moment axioms the word axiom postulates etc they are one of the same Euclid has used this particular kind of phrase postulates which more or less means the same thing as axiom that is what we are trying to use in the modern mathematics in particular. So an axiom is considered to be a statement usually a statement is considered by the true or false that is assumed to be true without any proof and all it doesn't require any proof for example we will be presenting some this axiomatic system Russell and Whitehead axiomatic system suppose if you says that these are my five axioms that means any one of these axioms cannot be deduced from any other axiom that means one axioms should not reduce to another one so it so happened that Paul Bernes the logician which was come later contemporary to better Russell and Whitehead he came up with the proof of his fifth axiom of Russell Whitehead axiomatic system can be deduced from other axioms in that sense it loses its status of axiom it will become a theorem so when Russell and Whitehead formulated is axiomatic system it has five axioms. Suppose if it is proved any axiom is proved from any other thing then it is considered to be a theorem it will lose this axiomatic status so coming up with this axioms is the most difficult kind of thing so axiom is a sentence which doesn't require any proof you are like silk evident kind of truths in order to start any branch any field of enquiry as you are talking about physics or mathematics or anything one needs to start with some kind of metaphysical assumptions for example when you are studying physics in particular concept of matter and all these things you will be asking several questions in all how the matter has come into existence then you will date you will go back to some billions of years ago and then you start saying that there was a big bang and then from then due to the big bang then it has resulted in some kind of hot planets etcetera hot things and all it started cooling cooling and all and different planets have formed etcetera our universe has originated by means of big bang so now if you go back and ask how this big bang took place and all you are all silent and all these cases so we assume that there was a big bang and then from that all the universe has originated so in the same way when you are talking about theology or something like that you will have your metaphysical assumptions such as God has created this universe in six days and seventh day you took rest etcetera so you need to start with some kind of metaphysical assumptions to do anything to start any kind of enquiry and all so in the same way the axiomatic system you need to start with axioms which doesn't require any proof and all they are self evident kind of statements which are obviously true it's like 2 plus 2 is equal to 4 which cannot be questioned and all of course one can even question the foundations of any area and all you can start questioning the axioms but usually it is the case that they are assumed to be true without any proof so these are also considered to be basic building blocks from which all the other things can be constructed just like you are constructing a building you need to have bricks and bricks are arranged in certain way and you will build nice structures. Examples could be Euclid's five postulates general of frankle axioms or pianos axioms in arithmetic etcetera general of frankle axioms in the set theme so now what you mean by a theorem a theorem is a mathematical statement that is proved using some kind of rigorous mathematical reasoning that is what we spoke about it in extensive detail that is a deduction a deduction is considered to be a deductive argument is considered to be an argument in which if your premises are in certain they are absolutely true etcetera so that is what we mean by deduction in a mathematical any mathematical paper that you come across the term theorem is often reserved for the most important results if you consider any mathematical research paper and all is often considered as one of the important results that is considered to be theorem and that needs to have a proof in mathematics if you say that I have come up with a theorem and then if I do not show I mean that it is it can be proved in all that will never serve as a theorem you need to provide a rigorous proof for whatever result that you are trying to talk about in a mathematical paper so in a nutshell the last step of your proof is what is considered to be a true statement that is considered to be a theory a theorem so now most of the time we will be using these words lemma whenever you are reading some kind of mathematical mathematics books and all you will be seeing all these axioms theorems lemmas corollaries etcetera what do you mean by lemma it is a minor result whose sole purpose is to help in proving a particular kind of theorem they are supporting kind of theorems so it is a stepping stone on the path of proving certain kind of theorems is a minor kind of important steps that you will be taking up in proving the major theorems so that is what we call it as a lemma and there are some other terms which are also very important so they are like this so what these are the terms which you commonly come across when you are reading a mathematical paper or any other mathematical text book a corollary a corollary is an important result which the proof relies heavily on a given theorem so we often say that this is a corollary of a theorem a etcetera and all this is some kind of thing which comes as an outcome of your particular kind of theorem so this is what is called as a corollary and proposition is the one which we have been discussing right from the beginning of this course that is any sentence which is usually considered to be as a true or false that is called as a preposition so it can also in the context of mathematics it is a proved and often interesting result but generally less important than a theorem a theorem is considered to be the most rigorous kind of a statement and all in your enquiry and of course axioms are all which doesn't require any proof and all they are absolutely true and all which doesn't require any proofs so now the other term which you come across is the conjecture so conjecture is considered to be a statement that is said to be proved and all it is unproved as of now but it is believed to be true so there are many such conjectures which are our gut feeling says that they are obviously true and all they are all true statements but as of now they didn't find any proof in the same way when Fermat has proposed Fermat's last theorem it took almost 200 to 300 years to prove Fermat's last theorem and all till that time it remained as a conjecture till somebody some mathematician has come up with a rigorous proof for Fermat's last theorem conjectures are yet to be proved kind of theorems and all we are our gut feelings tells us that they are obviously true but still it didn't find any won't be in a position to find a rigorous proof for that so this is what we called as a conjecture so all these conjectures can be refuted or maybe it can be proved also so till we find a rigorous proof for this conjecture we will not accept it as a theorem but all conjectures may be potentially capable of becoming theorems once it finds a rigorous proof the last thing that we will be using in the in this context is the claim a claim is an assertion in any mathematical paper you will be coming across this particular kind of thing so in any research paper etc and all you will be claiming something so that is considered to be an assertion that is then obviously eventually it is going to be proved it is slightly different from the conjecture conjecture is the one which our gut feeling tells us that it is obviously true it is often used like some kind of informal lemma it is not lemma as such but it is considered to be a kind of informal kind of lemma so that is what is called as a claim so now given this set of keywords etc and all what is that we are essentially trying to do is that we are presenting one particular kind of axiomatic system which is in which the proofs are considered to be rigorous as well as there are no other kind of hidden kind of assumptions which will become part and parcel of your proof I mean just as in the case of Euclidean axioms there are certain extra assumptions etc which has come which which also played a crucial role in formulating the proofs within the geometry we want to be we want to avoid such kind of thing so our proof have proofs have to be rigorous so what do you mean by a proof so now we have axioms theorems lemmas corollaries conjectures etc and all claims prepositions now with using these things what do you mean by a proof so usually a proof is considered to be a sequence of formulas and all these formulas will find some kind of justification so each step needs to be justified by some kind of a statement that is either whether you used modus ponens rule or use transformation rule or it has come from axiom or it has it is just another kind of theorem all these things needs to be addressed so each line of the proof in axiomatic system L it should have this particular kind of things so in a proof either the particular kind of thing has to be an axiom that means it does not need any proof and all you just write it and end it like that you say that that is an axiom that itself is a since there is not require any proof and all so that itself will be some kind of thesis and all thesis is the one which we have introduced earlier in that context we say that it is a theorem it is as well as an axiom also because the last step of your proof is considered to be a theorem so if you are given axiom and all you don't it won't require any proof and all then obviously it is considered to be a thesis so that is an exceptional kind of thing and all it's obviously a tautology etc and all so an axiom doesn't require any proof suppose if we come up with the proof of that one it will lose its axiomatic status so that is the first thing that you need to have and the second thing is that the next step whatever step that you are trying to consider whether or not it is a result of applying some kind of transformation applying some kind of rules of influence one rule of influence that we will be commonly using is the modus ponens rule that is a a and b follows from that particular kind of or you can begin your proof by assuming some kind of thing like it can be considered as an hypothesis which is already a part of you the given formula that needs to be proved so we discussed in the context of natural reduction method suppose if you are trying to prove a in plus b b in plus c and a in plus c so now what you will do is you list out all the hypothesis that is a in plus b b in plus c and in addition to that you will assume the antecedent of the conclusion that is a and from these three whether or not c can be deduced is the one which we are trying to see either it should be a hypothesis which comes from the given formula or it must be some kind of a lemma which is supporting kind of theorem it's also considered to be true so our journey starts from truth that is a axiom and transfer apply transformation rules to it is also true and then we are using truth preserving rules such as modus ponens etc and all it's a deductive kind of reasoning so that is also obviously absolutely true etc every step is considered to be truth so that's why our journey ends with the final step of your proof that is considered to be a theorem so the last step of your proof is usually considered to be a theorem so you need to note that one can come to the destination in finite steps and all some sometimes you can come up with the 15 steps to find the final kind of reach the destination that is the theorem or sometimes you might prove the same thing within some 10 steps and all so the proof which consists of cancer is obviously considered to be a rigorous kind of effective kind of proof when compared to a proof which consists of 15 steps are maybe more than that so as far as possible or in excessive information should not be there to maintain informational economy and in order to make this proofs effective your proof has to be the steps of your proof has to be as minimal as possible so when we write this thing l single term style a that means a is considered to be deduced from l since a is considered to be the last step of your proof so a is called as a theorem so that means we write z1 z2 zn etc the conjunction of all these formulas usually it can be it can be called as premises etc and a is considered to be the conclusion and that is deduced within the formal system l if and only if if a finds a proof in a given formal system from a given set of formulas z1 z2 zn etc it says that if we have a proof of for this thing q from p then we are guaranteed that we have a proof of p plus q as well so this is a sort of deduction theorem which was later introduced by her brand etc and all so we will talk about this particular kind of deduction theorem a little bit later so what essentially we are trying to do is from q if q is obtained from p then you discharge your assumptions like pq etc and all and you will start talking about p implies q rather than just p and just q so now there are in the elementary I mean suppose if you go back to our elementary schooling etc and all in the in our childhood we might have derived certain proofs which are considered to be bogus proofs but it might have convinced us at that stage so let us assuming let us assume that whether or not this constitute to be a genuine proof or it is a bogus proof or it requires some kind of fixing etc and all fixing the problem that arises in one of this steps in all so what is considered to be a proof so far we have discussed that each step has to be true and it should find a justification if a justification is wrong then you cannot move to the further step so there means some this proof is considered to be defective or there seems to be some problem with the proof so now observe this particular kind of proof to is equivalent to one so for that you start with some assumptions let us assume that a is equivalent to b then you multiply a both sides and the second step you will get a square is equal to a b now you add a square to both the both sides and all LHS and RHS then it will become the third step that is a square plus a square is equal to a square plus a b because you have added a square to both sides LHS and RHS so now the in the step 3 LHS will become 2 a square equivalent to a square plus a b so now you subtract minus 2 a b both sides this is what you get so that is 2 a square minus the six step 2 a square minus 2 a b and a square minus a b so now you take two common of these things 2 into a square minus a b and 1 into a square minus a b so it is like 2 into a square minus a b is equal to 1 into a square minus a b so this is what you got it from a is equal to b all the way down you got this particular kind of step till here it does not seem to be a little bit problematic in all because you started with all these assumptions etc and all they are not just done anything etc except that we are added subtracting from both sides etc so now the problem real real problem comes from is that the moment you cancel these two things then suppose if you say that so this is the n minus one step and this is the nth step you cancel a square minus a b both sides and then you will infer that 2 is equal to 1 and you can say that a square minus a b and a square minus a b it will become 2 is equal to 1 but one of the important principles in the logic is that you cannot cancel in such a way that you know a square minus a b from both sides because when especially when a is equal to b then a square minus a b usually it will become so when a is equivalent to b it is substitute for a b and all becomes b square because it is b into b is b so now it will become 0 so 0 cannot be cancelled with another 0 and all so it is in that sense our principles of mathematics will not permit us to move from this it will not allow us to cancel this particular kind of step and all so now this will not lead us to the next step because there was some problem with the justification our justification is defective and all because 0 cannot be cancelled by 0 so that is not permitted in at least in the principles of arithmetic so that will not let us allow that 2 is equal to 1 of course if you do not notice it properly and all sideline this particular kind of thing it appears that you know it appears to be a wonderful or nice proof and all but you can you can stop it step number 7 and you can question how did you cancel a square minus a b both sides and all 0 cannot be cancelled by 0 0 cannot be cancelled by 0 so this step is not allowed so that is the reason why this itself the proof itself starts stops here and then you need not have to talk about this particular kind of so now each step of your proof needs to be justified so here there was a problem with the justification it goes against the principles of arithmetic so you will not generate a proof for 2 is equal to 1 if you generate a proof for 2 is equal to 1 and then say that this is your effective proof and all and this is not considered to be an effective proof it is a bogus kind of proof so then the next question that comes to us is that what constitutes an effective proof so as it appears that from this particular kind of thing what we can learn from this particular kind of proof is that if your proof has to be effective etc and all each step should find a justification that should be according to the principles of logic etc in the same way when you are talking about axiomatic system which is based on set of axioms and transformation rules etc that thing has to when each step needs to be justified either by using the axiom or transformation rules applied on axiom or it should be it should result in from by up in from applying some kind of modus ponens principle or I mean any one of these things should be there then only the next step is going to be justified but here the seventh step is a problematic step your proof ends there itself and you can you can clearly say that 2 cannot be equivalent to 1 so there are certain things which are very important mathematicians would always be interested in starting with the tautologies that means the true statement a mathematician will never begin with contradictions the contradiction is a one sentence which can be spoken as both true and both false and x and not x is kind of inconsistent statement is unsatisfiable also considered to be contradictory to each other p and not p is contradictory to each other so why mathematicians think that this the contradictions are going to be considered to be as a hell so here is one of the interesting and funny kind of example which is given by a famous mathematician again they can't since we are talking about Russell white had axiomatic system so this example is also due to Russell invited Russell Betten Russell in one of the parties which he attended he funnily proved that a contradiction implies anything so that is the reason why you know one mathematicians will always be loving this tautology is etc and all from truths you generate truths etc and all but when you start with the contradiction you can derive anything so this is what is the thing he just gives a funny kind of proof so that is like this is what Betten Russell says if 2 plus 2 is equal to 5 that is a false statement he false statement implies anything then he is trying to prove any strange kind of proposition that you know that he is going to be a pope he wants to establish himself that if 2 plus is going to be 5 then he is going to establish that I am going to be a pope I mean in Betten Russell here so now so these are this is the proof that he tries to give is a funny kind of proof and all this not be taken very seriously and all but usually mathematicians would love to start with truth tautologies are rather than the contradictions because contradiction needs to help so now the proof goes like this if 2 plus 2 is equal to 5 then if you subtract 3 from both sides so that is like this first you take the antecedent of this particular kind of thing so now our assumption is that 2 plus 2 is equal to 5 so now 2 plus 2 minus 3 you substituted minus 3 from both sides subtract sorry not substitutes subtract 3 from both sides so this is what you get so now 4 minus 3 is 1 and this is equivalent to 2 so this is the proof and all so this is your hypothesis 2 plus 2 is equal to 5 you are assuming that that is true if that is true then the next step that is 2 plus 2 minus 3 5 minus 3 also has to be true that means 1 is 2 is also true now the third step is that is funnily he has used it is not a rigorous kind of proof and all it is only is this proof is only used for the sake of fun so now better Russell says that better Russell and Pope are two different people better Russell suppose one is taken as a better Russell and two means that better Russell and Pope are two different things so now this equation tells us that better Russell and better Russell and Pope viewed it in different way they are one of the same since one is already equivalent to better Russell better Russell and Pope also has to be same even if there is another person exist in all Pope the two has to be equivalent to one so it is in that sense better Russell and Pope are considered to be one person so that means he has to be none other than Pope himself that means he proved that he is the Pope so this is a funny kind of example in all so which this is come from this thing that falsity implies anything so this the same thing which can be proved in classical logic in a different way by using the principles of logic so that is the reason why we do not begin with contradictions in a given proof for example you start with P and not so this is we know that it is an apparent contradiction a statement cannot be both true and both false so now this is what is given if you assume that this is true and even these are also true because of this particular kind of principles so this eliminate this conjunction then P will become true and in P and not P not P is also going to be true so what we have done is we have eliminated this conjunction so this is what is called as impossibility or something which is a contradiction so now fourth one since not P is already true is also always considered to be true then irrespective of whether or not Q is true or Q is false the whole statement is going to be true because of this particular kind of formula this we know that the semantics of disjunction is like this that if it is F and F T F T F and all then it is going to be false only in this case in all of the cases is going to be true so now we know that this is already true so that means these are the two things which you need to take into consideration instead of P we have not P here so now irrespective of whether Q is P whether Q is false the P or Q is going to be true only so in that sense this statement is also true so now in the fifth step you have P here not P or Q now so these two so how did we come to this one this is law of addition so now this and this 3 and 4 sorry 2 and 4 disjunctive syllogism leads to this one Q so that means any particular kind of strange preposition one can prove so starting from the contradiction suppose if you say that it is raining and it is not raining then the statement is going to be pigs flies and all Q is considered to be pigs flies so this is one way of proving this thing so now the problem here is that you can prove either Q and even you can even prove not Q also so how can you get this not P again the same proof first step you take the same assumptions and 4 so instead of this one what we try to do is since P is already true you can add any strange kind of preposition Q and this is also going to be true so now fifth one so these this is law of addition this is the case of this particular kind of thing so now observe these two 3 and 4 disjunctive syllogism will lead to not Q so now with the contradiction you proved not Q and you with the contradiction even you proved Q and all so that makes your system inconsistent since you are derived Q and not Q is part of your formal system whatever system you are trying to talk about L so it is in that sense your formal logical system is going to be inconsistent that is what we are trying to avoid so this eventually led to different kinds of theorems which are little bit counterintuitive and all like paradox of material implication which we talk about it after we introduce Russell whitehead axiomatic system so now with this particular kind of note I will try to end it and the all other proofs etc we will try to do it in the next lecture so now what is that Russell and whitehead as achieved in the book principia mathematical lots of things which he did it is a voluminous kind of work and all which consists of three books ranging from 400 to 500 pages each book has 400 pages so now better Russell and white in the book principia mathematical which is considered to be a path breaking kind of book one of the greatest books of this 19th 20th century in that it is a kind of some kind of constructivist project constructed proofs etc it suit to show that all arithmetic can be reduced to logic that means there are two things which are important here when you whenever you talk about some statement in the arithmetic that can find an appropriate translation in the logic that means if you if you utter any statement in the arithmetic it will have its corresponding language you can discuss everything with the help of only this axioms transformation rules and other things so this grand program is what is considered to be a program which is called as the logistic so logisticism is like this so the thesis of logisticism is that all the mathematical concepts are definable in terms of logical concepts so that means you express any kind of plus operation or anything in arithmetic etc that has its corresponding logical operation that means it is in that sense you can talk all the mathematical truths in terms of logical truths logical truths are axioms etc and all so any kind of mathematical statement can be converted into an appropriate appropriately into a corresponding logical statement and all the mathematical modes of inference are reduced to logical mode of modes of inference and it is in that sense all mathematical knowledge is nothing but a real logical knowledge so so the basic idea here is that mathematics is a branch of logic or all the mathematical concepts can be reduced into the concepts of logic for example if you if you come up with an axiomatic system which consists of only a set of axioms and transformation rules and the rules of inference as minimal as possible rules are as minimal as possible and then you start talking everything in the language of this axiomatic system so what is there in our language of axiomatic system with respect to principia mathematical we have some Russell invited has five axioms and then there is a transformation rules and substitution rules and then the modus ponens principle that's it that's all we have and all the truths of mathematical arithmetic etc they can be appropriately translated into this one of this axioms all the p in plus q in plus etc will become statements of arithmetic and if we can do that particular kind of thing then it attains is rigor and all because whenever you are proving certain kind of theorem such as simple theorems like p in plus p or law of excluded middle etc and all that can be this p skews ours can be some of the statements in mathematics so now we are generating all the mathematical theorems we are trying to show that they will come as logical theorems now so mathematical knowledge will now turns out to be a logical knowledge because we are using only axioms and modus ponens etc and all so this is what we do in the principia mathematical so mathematics can be appropriately reduced to logic so now there are other things which Russell invited is trying to achieve the other major historical figures in the constructivist camp so how did they construct they constructed various kinds of theorems based on few set of rules which are few set of axioms rules and the transformation substitution rules there are lots of other axiomatic systems which are already there before Russell invited you have freguese axiomatic system in canter and of course after that David Hilbert were trying to axiometric geometry and he has come up with axiomatic system David Hilbert axiomatic system and Paul Bernays especially showed that one of the axioms of Russell invited can no longer have that particular kind of status we have different axiomatic systems and Pano Sardimetic also comes under this particular kind of category all the mathematics should be developed through appropriate definitions in the systems of logic defined in the principia if you can achieve that particular kind of task and all then you are said to have reduced mathematics to logic in the sense mathematics is considered to be a branch of logic so all the arithmetic analysis the other things which you commonly come across in the mathematics set 3D etc if this can be reduced to a set of concepts of logic then it can be called as developments of pure logic rather than the development of mathematics so in the next class what we will be doing is we will be focusing our attention on the main book the principia of Mathematica due to better Russell invited where there is a chapter specific chapter on deduction where using his set of axioms set of axioms he showed that for example law of contradiction law of excluded middle law of identity etc can be derived from this set of axioms so you should note that we should note that any formal axiomatic system that you are going to come up with at least we have to ensure that one should be in a position to derive the minimal things such as law of excluded middle that is a sentence is either true or false or law of identity such as P implies P etc all these things should come as theorems in your axiomatic system in the next class using the principles using the axioms of Russell whitehead will be proving certain important theorems such as law of identity law of excluded middle law of contra position all these things are valid theorems so now what is that we are trying to achieve in a nutshell is that we know that certain kinds of valid formulas exist in our formal logical system so now we are trying to find a rigorous proof for this valid formulas valid well-formed formulas so that is what we are trying to achieve and this will constitute a part of the proof theory so once we introduce this axiomatic systems then we will discuss about whether or not your when a given axiomatic system is going to be consistent that means when you will be in a position whether or not you are in a position to derive both x and not x if you are if you are able to derive x and not x as your theorem your axiomatic system that axiomatic system is going to be inconsistent or we will be talking about all the valid formulas whether it finds proof etc and all so that is all the two statements are provable and all the provable statements are true that is a soundness property or whether you are a formal axiomatic system is going to be complete etc and what are the limitations etc all these things we will discuss in the forthcoming classes.


It is really important that we do not conduct a trial by media or a trial by the public. This is a decision for the police to make on the facts and they will have best access to the facts and I'm leaving it in their hands. That is not a decision that should be informed by pressure from politicians or from the media. I've never tripped up, I've never stuffed up, I've done the job.


Stevenson defeats Mary Washington 62 60 tonight Glenn Clark with Morgan to bury 11 points tonight Morgan Let me just take me through you knew you were facing one of the best teams in the country tonight What was the conversation like with this team the last couple days? What made you guys believe that you could do this? Honestly our bench just we kept saying put bring energy on the bench And that's what helped us keep our momentum in the game like we were with them neck and neck throughout the entire game from the beginning So that was very evident when when Kelly scored at the end I'm pretty sure the entire bench was actually inside the three-point line at that Pretty emotional As the game went along, you know, like did you feel it? Did you feel the entire time like I feel like everyone felt it The entire time that was evident You that drive late in the game take me through that and Just what was going through your mind in those situations now in every possession was critical Success It didn't get blocked it was a huge bucket late what this does for this team now We're just talking about this with coach you obviously this wasn't individually your goal Right you have a lot to go in the conference But what this does the rest of the way to win a game against a team like this How much can you build off of this we can build off so much just next game on Wednesday? Just keep up the same energy honestly. All right, Morgan. Who do we need to shout out who do you need to say hello to my mom? Morgan congratulations.   Thanks for taking the time for you. Thank you Morgan DeBerry, I'm Glenn Clark. Go Mustang sports comm


Our agri-food systems continue facing challenges to provide enough nutritious food for a growing world population. Innovation and digital services are important driving forces for the agri-food development and the rural transformation. They create new opportunities for farmers and the rural people. Small island developing states see share a unique side of challenges. They face frequent natural disasters, they limit the arboline, and they depend on small-scale culture and the higher-priced inputs. The good news is that digital technologies offer promised opportunities to improve food production and trade, especially to small-holder farmers. They can have achieved food and nutrition security and global poverty alleviation, including for the seeds.


So on the latest data, renters have just copped the biggest rent increases in a quarter, three months, in 17 years. House rents have gone up by over 5%. At the same time, the latest wages data, wages have only gone up 0.9%.  The average median rent now across the country, $630. That means even if you're earning $105,000 a year on average, in an average rental, you're going to be in rental stress. Now, what does this practically mean? Single moms choosing between feeding their kids and paying the rent. Young couples being evicted, sometimes onto the street or back to their parents' place, giving up on ever being able to afford to buy a home, let alone cover the massive rent increases. We know how to stop this. In very practical terms, what the government can do is freeze and cap rent increases. In fact, if the government and Labor had listened to us last year and introduced a two-year freeze on rent increases and cap rents after that, then how many people could we have stopped being evicted onto the streets? How many rent increases could we have stopped? How many thousands of dollars a year in rent could we have same for millions of renters across the country? The bottom line is we have a housing system stacked in favour of property investors, property developers and the banks, and today's data shows the human consequences of that because behind every single rent increase is a story from a renter of doing it tough, making sacrifices, being evicted onto the street, sometimes onto homelessness. If you're losing a bit of hope, though, at the moment, the bottom line is this, one-third of the country rents. We have an enormously powerful political voice and increasingly it is going to be on us collectively demanding of the government real action that is going to win a freeze and cap on rent increases.


Alright guys, so we're gonna go over the recent code forces problem. This is the division to a problem so you're given the values a b c d and the condition basically of the problem is just that um the problem is really really boils down to this let me just write these write it down so the problem just boils down to this which is the does there exist a condition given a b c d where both of these are true right the um all the values of xi which is just like any of the any value you choose from is between we know that's between a minus b and a plus b so our does there exist values such that both of these statements are true okay so pretty much given like given a b c d so to do this it's actually not that hard first of all let's let's just rewrite this summation of n from i equals 1 to n of xi and we know xi is constant it's like a regular constant because like we know we're given this right we're given this xi it's like any any constant that you have you're just multiplying it so so let's let's formulate what this xi is so if um from your math equations you know that if xi is a constant we know that we could pretty much just take the xi out and just multiply n times xi right because this is just summing up 1 to n of xi every single time so then we could just multiply 1 to n by xi right this summation that's what it that's what the summation is so now that we have this summation let's just factor out the equation of this so in let's think about this pretty much in what scenario can both of these statements not be true right what scenario can both of these statements not be true right so we know that a b c d are integers right and we need to make sure that both of these statements are true right so and xi is just like any any value uh any index of each grain right any grain so uh pretty much the only way this condition could be false both of these could be false is if a minus b summation of a minus b is actually greater than the upper bound as possible or the upper bound of a plus b is actually less than the lower bound of c minus d right so let's just sum that up let's let's calculate the upper and lower bound of a minus b and a plus b so let's do summation of i equal 1 to n of a minus b so this is a lower bound we find the lower bound it's just n times a minus b and then if we find the summation of the upper bound it's just equal to n times a plus b right so we know the now we know the equations of n and a both of the upper the lower bound and the upper bound so this is lower and upper bound so after we know we know the equations of both of these all we have to do is just check if the this equation doesn't satisfy uh the this this equation so the case is where it's not possible is if the lower bound is greater than the upper bound right the lower bound of xi is greater than the upper bound the summation of xi of c plus d so that'll just be if if n times a minus b is greater than the upper bound of c plus d so that's not possible or the upper bound uh the upper bound of xi is smaller than c minus d and what's the upper bound of xi uh a plus b that's n times a a plus b right upper bound of xi and that's n times a plus b is uh less than the lower bound of c minus d so yeah so if any of these conditions are true then that both of these conditions are true that means it's not possible it's either this is not true or this is not true then it's not possible otherwise it's possible right that they're in the right ranges right so uh yeah that's pretty much how you do this problem this problem just boils down whether uh your values are in the right ranges okay you're just checking if the boundaries are in between it or not so that's the that's what this problem problem boils down to okay um yeah i'll just show you the code right now okay so this is the code um we'll read in n and then a b c d so if a minus b times n is greater than c plus d remember that's checking that condition that was not possible then or a plus b times n is less than c minus d then we see out no otherwise we print out yes yeah and this is my integer for t which is number of test cases while t minus minus so i'll just read in n a b c d okay so yeah that's all that you need to know for this problem uh it was just checking if the bounds are possible for the summation where x i is like any number between a plus b a minus b and b and a plus b so during if the if it's in the right range and uh if it's not in the right range then you print out no if it is in the right range yeah you just print out yes so yeah that's all you have to do rate com subscribe i haven't been uploading a lot lately because of school but i'll try to get i'll try to upload every week every friday so come back next friday all right guys so yeah uh i'll just see you guys later peace


Okay, we already have about 10 hand raises and we'll just go through one by one and yeah let's take a question from Padampad Singhanya University. Good evening ma'am. I'll really like your session. I have a question. We are teaching in private engineering colleges, private university and when we have a diversified students, students who has a, you know, good level of understanding, level of understanding which is very poor means when that students are diversified, you know, how to address those students. First, second, when you're talking about methodology adopted during teaching, I am actually teaching one of the course which is associated with our program that Cisco Networking Academy. There we have seen that we use a lot of simulation tool, a lot of games, computer games like Cisco has aspired games, our students designed some games. This kind of games also really help the students, you know, to understand the things better. I was going through, I teach networking, I was going through a paper in a published in IEEE Tripoli. They were showing that with a dancing model how, you know, people can understand WSI model better. One of my faculty also was there who actually understand this. One question I had in the morning, I could not ask Phataks that there was IIT Bombayx computer programming with MOOC model. We are really interested to join that, how to join that. This is one of the question was actually for Professor Phataks that if you can actually ask. And like online program and model, you know, I have seen a lot of companies are also conducting some tests. I was going through some material that Shibnathar University is conducting some tests through peers and you said conducting a test, judging a student, whether the student is good or bad or what kind of gradation to be there for the students is really challenging. Sometimes students, this kind of online test like Microsoft and Cisco, they use dumps and get good marks. Sometimes some good students get bad marks or average marks. So how to overcome such things with MOOC model? So these are my questions and these are my discussions. Thank you. Over to IIT Bombay. I am remote center coordinator. Thank you. Okay. Thank you. There were several questions here. Let's try to address them one by one. And some of them are actually quite interesting. The first two, let me mainly address the first two questions. And the first question was about student diversity. And I think this is something that all of us can identify with. Many teachers, almost all teachers can identify with various types of student diversity like starting from students, academic background and preparation to their motivation to just their level of engagement that they bring into their classes and so on. So what is it that we can do? And while this is a very common question, it's also a really challenging one in the sense that there is no magic bullet here. So I can't offer you a single unique answer. But what I can try is offer some suggestions as to what can be done to address diversity. And let's now, let's mainly take the diversity in terms of students academic background because motivational aspects is a completely different area and we can talk about that also. But in terms of students background, it really helps for the teacher to know what the student brings to the class, what motivation, why they're here in your class. And if you simply ask them, they'll say, well, it's a required class, it's a mandatory class and that's why I'm attending the class. But if we try to go one step beyond that as to what they want to do in terms of their career or what their profile of the class is, sometimes it might help to address real life examples tailored to the various backgrounds of the students. So you don't have to do each student every class but over say a month or so, if you're able to, if you know what their backgrounds and interests are and you're able to tailor some examples towards their interests, we've seen that students start paying attention because they feel that the teacher is paying attention to them. In terms of their actual diversity, the diversity in academic levels, is Bloom's taxonomy in fact gives us one framework for addressing it because what the taxonomy says is how do we set objectives and assessment questions at various levels. So our question papers are also have a variety in the levels. So students, all students should be able to answer at least some of the lower level questions and some questions can be set as being more challenging. So at least if you have a variety and diversity in your questions and the objectives in the class, you have some chance of addressing everybody. And finally, I think we have to somehow bring everybody at least to some standard that we think is required for the courses. So again, identifying which students are lagging and what to do for them, maybe some extra assignments for lagging students, maybe extra assignments for highly motivated students. All those are possible, more easy in small classes. In large classes, you can use learning management systems to do these additional assignments. But we have to make sure that every student feels that they are an integral part of the class and that we are doing something for them. Your second question was games. How do we use games in learning? What are the benefits? And I am glad to hear that you are using that a lot. Now, we have an entire session, I believe it is on day, one part of it is tomorrow and one part of it is on day 5 where we discuss how to use interactive visualizations like simulations or even games in an engineering curriculum. So you may get some ideas there on how to use them rather than simply showing it to students. And on the last day, we will also have a few sessions on how can you do some educational research in the sense, how can you measure what students are learning, whether their motivation improves if you use games. So there are ways to set up short studies and we will go over some research methods in educational technology. So that might help you answer this question on games and learning. The question for Professor Phatip, what I would suggest is just type it in Moodle and we will pass it on to him because he will be the best person to answer that. Let us take a question from Nyanamani College of Engineering. What about the rural background students, they are very weak in communication side. So we are facing the problem to teach in the technical oriented session. The language is the main drawback for our students. Even though our IIT people are giving a lot of options through off-line mode only. So this is a measure, this is a measure for our rural people. They are willing to study in engineering but the only thing is communication is very poor. Then one more thing, they are dominated by the township students. So what kind of means we want to improve to improve the quality of the rural background students. Again this is a very relevant and very challenging question on how to address the issue the drawback of communication, poor communication skills by some students. And I think I am going to first say that there are two separate issues. One is a language issue and a language barrier and one is an actual difficulty in communication. And the two are not necessarily the same. In the sense we have seen a lot of people who do not have language barrier and yet have poor communication skills or even if we allow people to speak and write in their mother tongue they are still not able to communicate. So these are separate issues. And regarding the language barrier I think the government is giving a lot of encouragement in creating ICT based materials in several Indian languages. So Professor Fatak mentioned spoken tutorials this morning and those have been dubbed into a variety of Indian languages. Many of the NPTEL videos also now are looking into aspects of translation and language barriers and how to overcome those. And about communication so ICT tools actually do have some stated benefits in helping students improve their communication skills. We will go into details of those in one session tomorrow called digital taxonomy for using ICT tools. But briefly what is done is students are given a lot of opportunity which is relevant to them in communicating. So communication is not just simply writing an answer on a test or asking a question. But for example what we tell students is that this is especially for third and fourth year students who are applying for internships and jobs we tell them that they have to create a portfolio of the projects that they have done to show their potential employer or when they go for an interview. So for the portfolio they have to create some way of presenting what all they have done. And because this assignment is extremely personally interesting for students it is their portfolio and their job interview. We have seen that a lot of people in fact do this seriously and this goes part way in addressing the communication skills. So collaboration tools that are or collaborative activities in class are also known to help improve students communication. Because it takes away the pressure of the teacher being there and the teacher telling them to do something. Instead they are just communicating with their friends but they are still talking about technical issues. So group projects where students are responsible for some specific deliverables structured in a particular way can help students negotiate the various roles that they play. And again we are going to do one session on group projects and collaboration and perhaps we will try to address communication also within these. Let us take a question from Sridhatha Institute of Engineering. Madam my question is which one is the best teaching learning methodology for the traditional one? We can go for peer instruction or we can go for say think, pair and share that Professor Pathak last time he has taught us in the workshop. So these are or any other method one related question is there that say suppose if we are introducing these changes whether the student community will accept those changes or to what extent they will accept that one madam. Ok. Over to you madam. My question the first part actually has no real answer. The first question was what is the best teaching methodology and there is no good answer because it depends on your goal the audience the logistical constraints you have and mainly your goal and even for a given goal there may be multiple methodologies that are valid. So I would suggest don't think in terms of which is the best methodology they really I mean that that's not a good question to ask instead a better question to ask is if this is our teaching learning goal what is a suitable methodology? So for example in today's peer instruction lab if you have gone through the activity constructor it says if you want to improve students conceptual understanding do this this and this. Tomorrow you will see statements like if you want students to design solutions to open ended problems TPS is a good solution. So sometimes direct instruction is also valid. So there really is no best methodology but having said that getting students to do something on their own getting beyond listening and reading has been known to be good for the most part. So active learning the reason we emphasize it so much is different types of active learning in fact can help different goals and a student possibly can be active even in a lecture class provided they have been given the opportunity to think of on the problem beforehand. So in terms of student acceptance it really depends on how we phrase it and how we frame it. Sometimes they find student resistance and at that point it always helps even before you encounter resistance it always helps to tell students why we are doing this and what are the benefits likely benefits. You may see a slightly slow acceptance in the beginning but if your students see that every third class there is one peer instruction question they will get used to it very soon. So if we present it in a positive manner and if many of your colleagues start using it the students will find that in every class in your college or in many classes they come across these active learning techniques and you won't find too much resistance in those cases. Let's go to Bhagwan Parushram Institute of Technology. Yes ma'am my question is that there are two students whose all India one is whose all India rank is in hundreds another in a normal engineering college whose rank is in lakhs but for both of them course curriculum and timings are same a very strong supporter of this ICT approach but in this short duration of time for the normal student it is quite difficult to complete the course contents in this time. So either there should be more time or there should be less contents so the question is you know what to do if we have to complete a syllabus in a given time and especially for some students or I would argue that for all students completing a given set syllabus in a given set time is not realistic and not even meaningful because what we are looking for is how much students have learnt we are so we should be less concerned about the breadth of the number of topics that we tell them so again this is a difficult question because of constraints if you did not have too many constraints so let's say you are an autonomous university you can decide along with your colleagues you can set your syllabus and curriculum in such a way that the number of topics that you deal with is manageable for your given audience ultimately what we want is students to learn deeply each topic to whatever level we specify if you think that you can give students extra time or extra time for assignments and all if you have that freedom do go ahead so in some institutes this is possible and I would urge all faculty to take advantage of this this flexibility of course you have to make sure that the following courses which depend on this course are not badly affected so that's why this is a dialogue that an entire department or an entire institute has to have together now if you don't have such flexibility let's say you are teaching something and the university paper setters set an exam then our hands are tied to a large extent and even then in most universities there is at least some part which is let's say 20% of the course grade is with the teacher and you can try these innovative teaching methods or deep learning deep teaching strategies in for that part if your students are willing to do some extra work that's also another option post them online you don't need to be with them face to face all the time so you have to exploit both the time and the flexibility to as much extent as you can to get your students but I mean this is really a sad problem and I think we all agree that it's not right to tell students that you have to learn this fixed amount and this fixed time I think professor Fartik is trying to make some policy changes also in this regard trying to give students get students to do the same amount in a larger time and so on okay GL Bajaj group of institutions good evening ma'am what are the research opportunity possible in education pedagogy for engineering background students okay so the question is what are the research opportunities for engineering background students in pedagogy or educational technology and I'm going to answer this slightly different question first and then I'll come to this question the slightly different question I'll pose first is what are the opportunities for engineering college teachers and the reason I pose this question is that each one of you is actually in a very unique position because all these new teaching methods that you're trying on your own or that you formally and systematically learn in this workshop and wish to try you have a lab at your disposal in the sense your class is your lab you can try some teaching strategy you can implement it and then you can actually do a study by gathering data systematically using it and also writing a research paper based on it we run a conference called technology for education it's an IEEE conference runs annually and we'll give you information about it later in the workshop so in that conference in fact we have several college teachers doing such studies and submitting papers it goes through a peer review of course but there is an opportunity now if you ask what are the opportunities for students in India it's still a little limited because there are a few places which have masters programs in educational technology where students can do such research there are even fewer IIT Bombay happens to be one of them places which have a PhD program in educational technology however there is a lot of opportunity in say the e-learning companies waiting to hire these graduates so many institutes now are thinking of starting MPEC programs in education technologies and I think the number is only going to grow in the next 5 or 10 years GF Gharda Institute of Technology Good evening madam this is Professor Bhartesh Danwade my question is that on PI basically this PI is used to check the correctness of the outcomes which the teacher has designed or it has some objective purpose to make the full proofing of the design of the the outcomes design an interesting question so let's explore the question is what is the role and purpose of peer instruction strategy is it to check if the teacher has taught such that all outcomes are met or is it for checking how many outcomes, how much outcomes the students have achieved and PI strategy actually has the advantage of doing both it is in a class of teaching strategies which are called as formative assessment and the word formative means is that the role of the assessment it's a question posed to students to think of it as an assessment but the role of the assessment is not to grade students and not to rank them not to say that these students are good and they pass and those students are poor but it is to get feedback both on the students learning and on the teachers teaching and act on the feedback so when we do two rounds of voting what students get is feedback on their own learning and their own understanding and their own misconceptions which they can correct right away at the same time as a teacher if I see that despite two rounds of voting most students happen to have the wrong answer wrong multiple choice it's feedback for me as a teacher that I didn't teach that unit properly and I have to do some other technique so peer instruction actually serves both these roles for the teacher and for the nada Neb Kushlenk engineering college Hello madam good evening Madam I'd like a clarification in Bloom's taxonomy we have create and apply so when we say when we ask the students to write a program that's a commander apply or create that is one question another question is what exactly is the difference between objective and outcome we say outcome based education and all what we normally do is that we give a set of objectives and then we also give a set of outcomes so what do you suggest madam but so far you have talked only about objectives and that of course comes like an outcome only these are my questions okay so first what is the difference between objectives and outcomes and the thing is we have only used the word objectives and why so and this is a slightly grey area in the sense that different people use different terminologies and different definitions so suppose there is some governing body like CTE or somebody who says that these are a list of objectives you must meet and these are a list of outcomes that is suppose somebody already gives you a definition please go with it okay because there are said this is really a matter of terminology between different groups different communities and so on the reason why we are not making a distinction is because at a conceptual level we consider both of them to be very similar to each other at most we can say whether the objectives are for your course and for a topic that I will teach tomorrow it is domain based and so on and the outcomes are more generic that students must develop the skill of teamwork so those can be put under outcomes some people make a distinction between domain based specific versus more genetic skill based but there is no really good unique accepted difference especially because these two are very similar in their purpose my suggestion to you is it does not really matter if you are not answerable to anybody for whether this is an objective or an outcome because finally we are looking at the learners performance and something that is measurable if on the other hand there is somebody who says hey here is a difference and you need to write both the objectives and the outcome then you go ahead with that definition it is not like textbook definition where something is very precisely written that this is the difference and the first question you ask the create versus supply so again you have to go back to seeing what the student needs to do so suppose we say we tell a student write a program to add the first ten integers so you have to think beforehand what all steps the student has to take mostly at a mental or cognitive level in order to answer this question at some level we are factoring in the student's try and knowledge and background because is it that we already have given them some sort of algorithm to doing it and they have to simply go and code that algorithm have we asked them to write the pseudo code beforehand so they know exactly the steps in that case it would become more of an apply level or on the other hand if we have to if the student has to think about some large system break it down make lots of decisions and write code for an entire to build an entire system the same act of writing a program now becomes a create level activity so it really depends on what are the cognitive tasks that a student has to do in order to apply or create it's not a single generic answer but what is it that the student is doing with respect to their prior knowledge cool of management sciences please go ahead my question is regarding active learning using peer instruction it is very unclear to us how we are going to complete it what it is aimed and how we are going to be the facilitator in the classroom for the student for this kind of teaching let me just try to understand your question are you asking how to complete the assignment or how to do it in class how we are going to complete the assignment and what are the roles we are going to play in the classroom for this kind of teaching methodology so a quick answer to the first part of the question how we are going to do the assignment there are three phases first is you understand what peer instruction is for that you go through the lesson there are videos and there are short questions you have to do that and then you will have an understanding of what peer instruction is and you already have been exposed to peer instruction during this workshop in the next phase what you have to do is you will have to construct your own peer instruction questions for that we have given a constructor an activity constructor we will also upload the pdf file that the activity constructor is referring to so use both of them to create peer instruction questions for your topic so you already started with learning objectives of a topic choose the same topic and see where you can introduce peer instruction so there are multiple types of peer instruction questions so choose any three of them and create complete the assignment so this is all regarding the assignment so regarding actual class implementation so assignment I think the deadline is about a week or so from now there was one announcement I had related to this so maybe this is a good time to say in the beginning around 330 we got a few questions as to what to do in the lab and essentially in each lab what you need to do is go to your lab and open Moodle and there will be a module called lab for day 1 p.m. 1 and it is going to be in order so just find the module for that particular day and that particular session click it and then there will be instructions within so these are carefully structured activities that are in Moodle today's lab assignment in order to complete it various parts of it you have another week to do it the other part of the question what to do as a teacher so peer instruction strategy for a teacher has two tasks really one task is to write the question suitable for a particular goal and the second task is to implement it in the classroom what the assignment today helps you do is to create various peer instruction questions for various teaching learning goals such as improving students conceptual understanding getting students to make to relate different representations such as a graph and any equation and so on once the question has been created by you for your class then when you are in your class with your students you can implement it in the same way that we did here you pose the multiple choice question let students vote get them to discuss maybe make them vote again give them hints on their reasoning the focus is on understanding and not on the right answer and finally summarize that activity so each PI question usually takes between 3 and let's say 7 minutes or so to complete it can be as short as 2 or 3 minutes also so this is part 2 of what you do as a teacher okay Saint Xavier's Catholic College of Engineering good evening madam our management is focusing 100% the result for the students so in this context we are not able to fix the setting the objectives exactly for the students so what is your suggestion in this kindly correct me if I am mistaken you wanted a strategy for 100% results in your college so when it takes as you told that is setting the objective on more in towards in the hierarchical way is preferable and will be effective and moving towards the 100% result your system or your suggestion may be ineffective but we cannot concentrate on the 100% result or more times only 100% result but as per your instruction and setting objectives first and moving towards the hierarchical way of preferences it will be less effective for the 100% result setting your courses or creating your course designing your course according to Revised Bloom's Taxonomy is not to get 100% result based on some arbitrary norms the goal of Revised Bloom's Taxonomy is so firstly it is that various tasks require different amounts of cognitive effort by students and as teachers we all intuitively and informally recognize that so this hierarchy tells us gives us a theoretical basis of how to how to tag a question how to label an objective as a lower thinking level or a higher thinking level what we did not discuss today is how many questions should be set at lower levels how many at higher levels and so on but our goal is that students might start at a lower level or in a mix of levels and we want them to at least go up the ladder we want some questions to be at the higher levels otherwise so if you only ask questions like define something and write a short note that is not engineering or science that is not learning so Bloom's Taxonomy help us define what it means to have meaningful learning in a given topic but it is not a formula for getting 100% result and yeah that is what I can say D.Y. Patil Institute of Engineering yeah this is Rashmi Deshpande my question is again regarding hierarchy of cognitive levels of learning objectives while deciding the parity we saw six levels of Bloom's Taxonomy and then in the activity that was conducted later we tried to assign one learning objective per level but what my understanding is in a single session it is not compulsory to have one learning objective per level I mean it depends on the content depends on the syllabus for some sessions we can have or say we can address three levels of Bloom's Taxonomy maybe in some other session we can access all the six of them is my understanding correct? yeah you are absolutely right so the comment here was that even though we have six levels within Bloom's Taxonomy it is not necessary that one session let's say one class or one unit has to address all six levels even though in the exercise we did it the reason we did it in the exercise is to see how much you can push the cognitive effort and thinking of a student without increasing the number of topics that was the purpose of the exercise but you are absolutely right within a given session it's up to the teachers it's up to your course design that you create is it a single level that you are targeting or is it two levels maybe it's levels one, three and five today and tomorrow it's only level four so all these combinations are possible but all these combinations are possible at the details level but if you take your entire semester or entire course what is recommended is that there is a variety of objectives at different levels so don't create a course and don't create an exam question paper which has only recall and understand level of questions that's a general recommendation but otherwise what you say is absolutely right Sagar institute of research and technology my question is in computer science area there are some subjects like compiler design or TOC theory of computation so we are finding difficulty in creating its learning objective as well as Bloom's taxonomy application is difficult in such numerical kind of subject so kindly suggest some methodology there are some subjects where it's difficult to write objectives at all Bloom's levels this is a question sometimes we get you also said that it's difficult to write learning objectives so that part is actually not valid in the sense whatever be your topic even if it's very descriptive there is always some expectations we have from a student and if you think about what you want to see the student do in front of you so imagine you are talking to a student and you are free to ask anything from the student and the student is free to show anything that you actually want the student to demonstrate to you that is the learning objective so it's not a valid statement to say that it's not possible to write learning objectives now in courses where there are calculations and applications of rules it should get so you said it's numerical my understanding is that there is some rule which is then implemented into some numerical equations in such courses it should be even more easy because you want students to demonstrate the application to calculate something numerical if so the first part is learning objectives are always possible to write and the way you do it is actually just asking yourself the question what do I want my student to be able to demonstrate in front of me after say one semester of this course sometimes people say that it's not possible to write learning objectives in all six levels and even there most of the times we have found that with a little bit of thinking it is possible so once concrete practical suggestion I have is look for use google and see if people have actually done this course in terms of learning objectives these days when since outcome based education is becoming very popular a lot of courses are being rewritten in terms of course not the course content but the course design and the course outlines are being rewritten in terms of learning objectives and see if anybody has created learning objectives for this course Rangasamy College of Technology good evening ma'am sir this is Ramesh from KAS Rangasamy College of Technology ok madam we are teaching different type of problematic objectives to our student so which ICT tools can be effectively used for simplified learning to our student civil engineering civil engineering yeah so the use of ICT tools actually is is not restricted to a particular domain and one reason we are using more examples from computer science and electronics here is because of the demographics we have among the participants a large number of participants are from these domains and that is why our expertise is also in these domains and that is why we are using these but use of ICT tools creating learning objectives this is possible in any domain so what we can do right now during this workshop is and look for examples that we have in terms of learning objectives and assessment questions from various domains that we have already created and we will pose them as examples in Moodle and in terms of using ICT tools you should look at the purpose of the tool so if you look at peer instruction and especially if you are going to use clickers the interesting or the relevant part is to create a clicker question this peer instruction question that you can use and in tomorrow's digital looms taxonomy session we will see that a lot of these tools can be used in any domain what is important is the instructional strategy and the task or activity that students do with this tool it is not the tool itself that is restrictive to the domain knowledge institute of technology good evening madam I am Prabhakaran faculty of mechanical engineering of knowledge institute of technology I have a one day one question is it possible to set objectives which includes all cognitive levels for an it is possible to set objectives at all levels within an undergraduate program the short answer is yes and I elaborate a little bit more most of the times it is possible to write objectives at all levels even within a single course even in introductory courses because the higher cognitive levels do not mean that we bring in more content or more sophisticated content it is that given a certain topic how what are the different levels of thinking that we want our students to do so for example in a first year first semester programming course you may think that design is out of the question but it is still possible for us to give students a project where they design a game using a drag and drop interface like scratch that is possible for even high school students or middle school students to do so possibility always exists and within the four year undergraduate program it is really up to you as a department to decide to what level you want to take each course so sometimes people might say that I only want to target some levels in the first part of the course in the first level of the course and next semester I want to target other levels that perhaps that can be done but try to address different levels within a single course Perumal the thing is that you have told about the ICT tools right so could you please give us the brief description about the ICT ICT tools so that it will be effective for all the teachers to use that in the classroom so ICT very broadly speaking is information and communication technology that includes almost every technology tool that we use today the most ubiquitous one being the computer and the internet what we will do in this workshop is give you exposure both from a learner's perspective and from the teacher's point of view to a sample of these tools so for example model that you use today is an ICT tool when you use avu avu is also an ICT tool avu pole can be considered as another tool google docs is an ICT tool so there are several hundreds if not thousands of ICT tools and we look at a few of these which have relevance in teaching and learning one common or very important and common tool that we will look at is interactive visualizations because in science and engineering topics it has an important role to play and there is no single listing available of all the ICT tools that exist but at the end of the workshop you will be able to make a list of some tools and some instructional purposes related to that Kurukshetri University my question is that to design this different methodologies like peer instruction TPS for a teacher's perspective it is okay the real problem comes when it comes to the motivation of the student like most of the time the students like which are taking admission in the engineering colleges the quest is not that like to learn engineering so how to keep motivate them throughout the four years that is one question the second question is like in a semester hardly you get 40 like 40 hours or 40 lectures per course so how to exactly manage the time while implementing these different teaching methodologies take the second question first because that may be a little easier to answer even though it does not have many of these questions you are asking are challenging and real in the sense that they do not have a textbook answer but it is still worth talking about it and the second question you asked was that we have 40 hours for a course and if we want you know how do we manage the time how do we do a break up of time and especially if we are trying some new teaching methodology to what how much weight should we give it so if you are using a new methodology we are not saying that you must use it in every class and you must use it for the entire class in fact that is something which is perhaps a bad idea but we are doing here is so if you notice any session especially in the next few days we will always start with and focus very heavily on the purpose of a particular activity or a task or a teaching methodology so if you know that the goal of this teaching method the intended goal the targeted improvement is for is something next time you are faced with a scenario where you have the same goal you can use this strategy so for example when we want to tomorrow when we do visualizations we will see that when you need to when students need to look at how different parts of a machine move visualizations are important so next time when you are doing some intricacies of the machine instead of explaining it on the blackboard you might choose you might choose to look for a video visualization so once you have a grip on for this goal this strategy is a good idea you can choose when to implement it and how much so I will give you examples of what people do here typically in every lecture we have at least one but usually about 2 active learning techniques some may be as small as a 3 minute peer instruction question sometimes it is a 15 minute think share question sometimes it is a 6 minute debate but there is between 1 and 2 active learning strategies that is what some instructors do some instructors entirely go the flipped classroom route which we will look at again so different models exist based on the instructors purposes and goals you are right that how do we manage to complete all the syllabus of active learning techniques and there is no good answer there either cut the content for which you are doing delivery that is possible because usually if you do one active learning strategy for a given content you do not students learn a lot more than just that content so they will be able to do self learning on some other parts of it first question you had on motivation of students that is in fact a more harder then I think I am only going to give a very personal answer here I think it is our responsibility on two fronts one is find out what students want to do they are all there in engineering because they think it will help them get the good jobs so let us try to relate it to what the workplace means that is at the second end so if at the workplace what the people who go to the new jobs if they are required to do group work and work on design thinking we should train them in those communication is also one important aspect so through our technical courses we should also look at some of these generic thinking skills and other such workplace needs you will find other people giving you lots of different answers all of these are valid because this is a very hard question but at least let us find out what is the motivation of students and it is okay if they do not want to learn Newton's third law or second law of thermodynamics Srinagarayana Guru Kulam College ma'am using the tools like this avu classroom the I think the very advantage is getting an expert from around the globe so have you like IIT the IIT people have initiated something not for the teachers teachers they have already a program for students and one more thing that is the situation every people will be going for the best teachers around the globe that may be a problem in the future let me just pass through your question and the point is with the availability of internet and avu and all these technologies there are a lot of content available and we think this is a good thing it is not a problem and the reason we think it is a good thing is that content is not the same as learning information is not the same as learning and content is not the same as learning so in the past days you used to have a library full of books and every teacher wrote another book and so on it is like now books and content is being created using technology media what is important now is to help our learners assimilate this information understand this content be able to judge which information can be used and applied in what way and that is the role of teachers today 30 years ago teachers were the only people who were experts and who had the information today information is there everywhere good information is there so let's our role has changed so there is no threat here at all the main advantage is we can break the barriers the geographical barriers so the objective of this workshop is like our students are there and we are very suppose we are the teachers from the campus only that is home tutors campus in the campus only so you will be discussing about the tools that can be used in the class itself for the better understanding so our focus is on use of tools for either better teaching or better learning these are two phases of the same coin our focus is not on just the tool itself but the point of why we are using the tool and how to use the tool for better teaching and learning that is going to be the focus of this entire workshop so we will select a few tools and we will look at several effective research based research proven effective instructional strategies in order to satisfy this goal so with this let's actually formally end today this has been a very interactive and interesting question and answer session I hope this was useful to you also ok thank you and good evening to all