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