 Definitely one question comes in J.E.Main from this chapter methodically. Now, in the era of geometry was evolving. I mean in the era of those new mathematicians coming up. Aristotle and all were working on geometry. There were a group of travelling teachers which they called for fists. Now these guys, they had a reputation of arguing for any point where they used some kind of logic to confuse people. For example, he'll say Laika is my cat, Laika is my mother, so my mother is a cat. They used to give some weird arguments where they would confuse people. So there were some kind of travelling teachers which we call as the sophists. I mean I'm just giving you a bit of background may not be very important for you. They were travelling teachers who used to give weird arguments to confuse people. And then Aristotle who was working on geometry at that time, he started writing a book called Refutation of the Sophists. That book was called Refutation of the Sophists. Where he started mentioning some kind of reasoning or logic in order to refute the argument given by the sophists. So what are we going to study today is basically logic which is a science of reasoning. So as you know in geometry we deal with points and lines and all those concepts. That time in logic we deal with statements. So first we need to understand what are our statements or what are our propositions. Have you done this chapter in school? Yes sir. Okay, so you tell me what is the meaning of a statement, a mathematical statement or a proposition? It's not vague. It's not vague. What about you Niyati? It makes complete sense. Okay, now there are three things normally we tend to misuse. One is utterance. Utterance is something which is just a verbal expression which may include nonsense expression. Like you know aprakad aprakad, something like this. It doesn't make any sense. Or let's say like goodly goodly good. So something which you say which doesn't have any meaning to it. That's one utterance. Second is a sentence. A sentence is basically those utterance which have a meaning. So utterance with a meaning is a sentence. Okay, so let's say like you know I give you eat your lunch. It's a sentence. Where were you going in the afternoon yesterday? Okay, give me a glass of water. Wishing you best of luck for your practical exams. These are all sentences. Now those sentences which can be answered as a true or a false but not both. That means there should not be any ambiguity whether the given sentence answer is true or a false. Then that will become a statement. Okay, to give you some example 2 plus 2 equals 4. The statement because you can say true for it. Why is the rational number? It's also a statement because you can say false for it. Yes or no? Yes. So things where you can conclude or sentences which you can conclude with a true or a false but not both are called statements. Now if I say raise your hand. Is this a statement? No. Why not? You can't say true for raise your hand. You can only do it. Yes, absolutely. May God bless you. Is it a sentence? Yes it is. Is it a statement? No it is not. No sir. Okay. All roses are white. Is it a statement? Yes sir. Yes. Okay. There is no time on this. These are something which you have already done in the school. So we represent a statement by a symbol P. P stands for proposition. And whenever we have to write a statement we write P column and then we write that statement. Right. So if I have to say a statement that Venkat is happy. Okay. I'll write it like this. Okay. Then there is something called denial of the statement or we call as the negation of the statement which we write as a symbol P preceded with a tilde sign. Okay. Yes sir. This is it. Okay. So the negative of this statement. So if I say Venkat is happy is a statement P. What is the negation of the statement P? Venkat is sad. Venkat is sad is not happy or it is not the case that Venkat is happy. So there are different versions of the same thing. You know you can write anything in the exam or it is false that Venkat is happy. All those things can be written for it. Okay. Now what type of questions can come on negation? There's something called quantifiers. I don't know whether you have done this in school. There are two types of quantifiers which we normally talk about when it's called a universal quantifier. And the other is called an existential quantifier. Okay. What are the universal quantifiers? Have you done this? No sir. The sentences which contain all or every like that are sentences which are said to contain universal quantifier. For example, let's say I give a statement all cats bite. So basically this statement is having a universal quantifier. Okay. All prime numbers are odd. It may not be correct. But I'm using all here to show a universal quantifier. Okay. Whereas statements which contains their exists or some. Okay. These are called existential quantifier. Right. So I'm sure you would have used this simple. For all. For all or their exists. Okay. These are basically representing universal and existential quantifiers. Now, why is this important is because whenever a sentence begins with universal quantifier, we have to be very careful while negating it. Let me give you a question. Let's say there is a statement that says all roses are white. Okay. What do you think is the negation of the statement? It's not the case that all roses are white. Wrong. Oh. There exists some rose that's not white. Right. Absolutely. So when you have a universal quantifier, remember its negation will start with or will contain an existential quantifier. So you can say there exists a rose which is not white or you can say it in various ways. Some roses are not white and not every rose is white. They are not all roses are white. Are you getting my point? So please note that never say the negation of a statement containing universal quantifier like all roses are white don't all roses are not white. Okay. Why sir? If you say all roses are not white, it will be marked wrong. Okay. Why? All roses are not white. Basically, if you are able to show an instance one of the rose which is not white, then you have negated the statement. You don't have to say all the roses are not white. For example, let's say everyone in France speaks French. I think you can join again. Is my voice echoing? No, it's nice. If it is echoing, I can hear the echo actually. Oh. If it is echoing, you can join again. There is no problem in joining again. Okay. Nice proper. I never felt any echo, sir. I'm using that thing, your phones. So let's say I give you everyone in France. I give a statement to everyone in French. What is the negation? There exists one guy in France who does not speak French. Will I say everyone in France doesn't speak French or all people in France are not speaking? That will be wrong, right? In order to negate this, even if I find one person who doesn't speak French, my job is done. Okay? Yes, sir. At least one person in France doesn't speak French. At least one person in France does not speak French. Or there exists a person in France who doesn't speak French. Or some people in France do not speak French. So there are various ways in which to negate it. Is that right? Sir, have a doubt. Yeah. Sir, when language is being created. So utterance, statement and sentence was all the same, sir. They were all the same. When it was being created, there was actually a meaning found out in the utterance. And then it was made a sentence. So at that time, for us now, Gouli Gouli didn't make any sense. Right. But at that time when it was being... Right. For you now, very German will not make sense. Right. So there is a pattern in which they are speaking it. So for them, it makes sense. But don't worry about it. We are dealing with... Okay, sir. Yeah. There is no way if a statement starts with an existential quantifier. Okay. Then what is the negation of that? It will start with a universal quantifier. For example, let's say there exists a cat. Which... There is glasses. Okay. Okay. What will be the negation of that? All cats bite. All cats bite. Is that fine? So remember one thing. Negation of a universal quantifier should begin with an existential quantifier. And negation of an existential quantifier must contain a universal quantifier. Okay. Let's see. Yes. We have a question based on this in our previous J exams. Okay. Yeah. Let's say I take this question. Question number five. I read this question. Let S be a non-empty subset of real numbers. This is the following statement. He says there is a rational number X belonging to the set S such that X is greater than zero. Which of the following statements is the negation of... C. Others, everybody has to participate. It's not like only one person. It appears no banquet. They have given the speaking role to you. Nobody is here other than you. Dheeraj was there. That also has stopped coming. I don't know why really. So Niyatti says something else. Lemker, you said C, right? Yes, sir. Okay. What about you, Amla? Adweta. Adweta. Say something. I talk a lot, sir. No, no, no. That's fine. Absolutely. You are at your normal behavior. Screen frozen. But mine is working, right? Read the question. The question says there is a rational number which is greater than zero. It starts with an existential quantifier. Isn't it? Yeah, I think C. That means you have an existential quantifier here. Then read carefully and tell me which of the options is correct. It should have a universal quantifier. Tautology. What's tautology? I'll come to that. Don't worry. Okay, sir. Tautology is a statement which is always true for whatever component values that whatever value the component statements have. Anyways, so in this case, can I say A is my answer? If I say X belongs to S, X is less than zero, implies X is not rational. Is this my answer? Yeah. Oh. No, I guess. No, sir. This is something else. There is a rational number X belonging to S says that X is less than equal to zero. Now existential quantifier negation cannot be another existential quantifier. So B is also wrong. Okay. Now there is no rational number such that X is less than equal to zero. Isn't it saying how is the negation of that? Yeah, this says here. Oh. There is no rational number such that X is less than equal to zero. That means they are saying all rational numbers are greater than zero, right? Hmm. Okay, so isn't it the same thing? Oh, yeah, yeah, yeah. Every rational number present in S is less than equal to zero is a negation of zero. So D is the correct answer. Oh, yeah. Niyati gave the right answer. Okay. So this is why you have to deal with this kind of no problem. And let me tell you one problem and definitely the next compound statements. Did I unshare this thing? No, sir. It's nice compound statements. A statement which contains more than one. Elementary statements. Those are called compound statements. Let me give a simple example. Let's say 18 is divisible by 6 and 3. Look at the statement. This is a basically statement made up of two component statements. And those statements are 18 is divisible by 6 and 18 is divisible by 3. And you have another statement. 18 is divisible by 3. And these two statements are being connected by a connection. It will be called here as the AND connector. Okay. Now this connector is basically represented by this symbol in logic. And such a statement will be called as a conjunction. Okay. So when you say p and q, mathematically or symbolically it is presented as p or paro, q. And the statement is called conjunction. And these two component statements are called conjuncts. These are called conjuncts. Okay. Now I'm sure you would have done Boolean logic in your computer science engineer classes. Correct? No sir. Oh. Anyways, Boolean logic is basically, you know, you can say a modern version of this logical reasoning that we had learned from our Greeks, which is applied to Boolean. George Boole was an English mathematician who applied that to Boolean logic and that is used mostly in electronic circuits. And it all did all those kind of things. Okay. The purpose of simplicity, you can assume it with a multiplication. Okay. Now how does a truth table of AND look like? There's something called truth table. What is a truth table? Yeah, 1100 and all right. So basically it is a kind of a tabulation of what is the outcome of a compound statement for different inputs of the component statement. For example, let's say if I say both my p and q are true, what can you say about p conjunction q? One. That will also be true. Yeah, one for you is true. Yes, sir. Okay. So what about this is true, this is false. This AND is false. Yes, because until I say 18 is divisible by 6 and 18 is divisible by 5. That means 18 is divisible by 6 and 5. You will immediately say false for this because it is not divisible by 5. Okay. If p is true, q is false, q is true, then also this is going to be false. And if both are false, then definitely it's going to be false. So the only occasion that your conjunction is going to be true when your conjuncts both are true. When your both the conjuncts are true, then also your conjunction will be true. Okay. Now let me tell you all statements containing AND should not be taken as a connector. Okay. For example, water and oil do not mix well. AND is not acting as a connector. Are you getting my point? Yes, sir. All the ANDs are connected. By the feel of the statement, you'll understand that it's acting as a connector or not. Okay. Now, next connector that we're going to talk about is basically a disjunction. Or we can say OR connector. OR connector also called as the disjunction. Representing the symbol down arrow. Let me give you an example. Let's say I give you a statement. Venkat is smart and lucky. Smart or lucky. Sorry. Okay. Now here you may be both. Yes, sir. There are two types of OR which we normally come across in our English language. One is called Exclusive OR. Another is called Inclusive OR. Ayoo. What is Inclusive OR? The statement that I gave you, this is actually an Inclusive OR. Because you can be both. What do you guys say? Nagma is dead or alive. Because he cannot be both at the same time. Okay. Or something like I will stay at home or I will go out to see a movie. That is an example. Home theater is not counted. Yes. So there you cannot improve both the situations in the statement. But here you can say yes. Venkat can be both smart and lucky. Okay. Unless there is something wrong with the bulb or with the wiring. Now both can happen. There may be something wrong with the bulb as well as the wiring. Okay. So that is Inclusive OR. So in our discussion, I will talk about Inclusive OR. I will not talk about Exclusive OR. Okay. Yes, sir. And again please repeat every OR as a connector. For example, I am going to watch the movie Seeta and Geeta. The OR is not connected. Okay. What about the truth table option? Let's talk about it. It will be true, true, true, false. False only when both are false. The true will give you a true. True false will give you again a true. False true will give you a true. And false false will give you a false. That means only when the component or the consequence of false then only your disjunction will be false. Yes, sir. Something very important here. What is the negation of disjunction? Sorry, conjunction. I claim that the negation of disjunction. The negation of the conjunction is basically disjunction of the negation. That means both are equivalent. Oh, the Morgan's law. This is actually the Morgan's law. Morgan's law. Can we prove this? In order to prove that any statement is equivalent to another or you can say this is the symbol of logical equivalence we just have to show that the truth table are the same. Okay. Can we now do a truth table of this guy and show that the truth table of these two matches. So what I'll do is I will make a truth table of this p, q, p and q. Sorry, before that I'll make not p, not q. Let's say not p, not q. Yes. I'll say I just got a crick in my leg. Okay. Not p and q. And finally we'll make it a not p or not q. Okay. And then check whether this column and this column matches or not. Let's take the perfect top. How many inputs do you have for p and q? Remember it's always two to the power of n. Two. Oh, that way. Yeah. So if there were p, q, r, let's say then it will have eight inputs. Okay. Let's say two, two. So what will be this? Hmm. This will be false. False. False or false, which is false. Correct. This will become false, true. False, true. False. False. It will become a true. This guy will be true. That will become a true. True or false, true. Yeah. False, true. True, false. True, false. True and false, false. False. False. False. Then true. True. Then this is also true. Correct. Yeah. Okay. False, false. Finally, let's talk about false, false. False, false. This will be true, true. True, true. False, true, true. False. Correct. No, no, no. And this will be true. Oh, yeah. Correct. So as you can see. Yes, sir. Two columns exactly match. That means these are logically equivalent to each other. A lot of questions will be asked. Which of the following statements are logically equivalent or not? Okay. So there, please remember to, you know, do it by truth stable. One sex. This is some one second copy. This is looking at you. I'll take screenshot. Yeah, yeah, please. Done, sir. Okay. In a similar way, falling. I can say. Negation of disjunction. Is logically equivalent to. Conjunction of their respective negations. Okay. I'll leave you as a person. Please prove it. Using. PT. Thank you for the exam. Is that fine? Yes, sir. Okay. Now comes the important part. So far we've been doing many activities. But now comes the serious part of this chapter. Which is called implications. Implications are also you can take some statements. But here. The connectors are not the and or the all. But it is something like this. Let me give you. Example of an implication. By the way, implication is also called conditional statements. Okay. Let me give you a. Typical example of. If it rains. Then I will not go out. Okay. As you can see, this is made up of. Two. Component statements. One is. It rains. It rains. It rains. It rains. It rains. It rains. It rains. It rains. It rains. Component statements. One is. It rains. And then I will not go out. Okay. So you can say. This is one statement. And this is another statement. Okay. And if. Then you symbolically represented as P. Single direction arrow. Okay. Where P is sometimes called the anti-city. Okay. Okay. Many books will call it as. Hypothesis. Many books will also call it as the. Reminds. Okay. Reminds. Oh. Q is called a. Consequence. Precedent. Oh. Consequence. Okay. Yes. Books will call it as the conclusion. Okay. Now. Why it is called a conditional statement because. For the second statement to happen or for the conclusion to happen. There is a precedent to that or anti-student to that. Okay. And remember. It is. The direction of the arrow is only one side. It doesn't go the other way. Right. What does it mean? It means that. What's that? I'll give you an example. Write it down. The fish in condition. Q. But q is a. Condition. What's that? I'll give you an example. I'll write it down. The fish in condition. The fish in condition. Q. But Q is a. Condition. What? T. T. Sufficient means it is. It is sufficient that if P happens. Then Q will happen. Right. But if Q happens. We may all. Are you getting my point? No sir. Let me give an example. If it is a Sunday. Okay. Then. School will be closed. So if somebody says to you that hey today is a Sunday. Then what? What conclusion can you draw about the school being open or closed? Pacta closed. It is definitely closed. Pacta closed. What is the consequence of that? School is closed. Does it mean it has to be a Sunday? Not necessarily. It may be closed because of some. Right. It may be closed because of some. You know. Right. It may be closed because of some. You know. Right in the town. Coronavirus. It may be closed because of Coronavirus. It may be closed because there is some leopard on the road. Are you getting my point? Yes sir. So this is a sufficient condition for this to happen. And this is a necessary condition for this to happen. Are you getting my point? Okay. Yes sir. It doesn't mean it is. Fulfilling the entire requirement. Are you getting my point? Yes sir. We can also read it as P implies Q. Okay. That's why it is called implication also. You can also read this as P implies Q. One second. One second. Sufficient. Have you done this in school? No sir. We spend like 35 minutes on this chapter. Okay. Okay. This is the most important part because a lot of questions will be framed on this. Even a final exam is coming for like one month. One to two months. We also one question will come. Not more than that. But there will be a short short question. We will definitely get four months. Yes sir. Okay. Yes sir. There is no need to write down. I will be sharing this with you. Seriously then chill. I am not writing. Enjoy this chapter. Okay sir. Now there are certain things which are associated with implications. So when you say P implies Q. Okay. This is an implication. There are certain terms that we need to understand. If you write this will talk. We call it as convert of this implication. Okay. Now remember. If P implies Q. That means this is true. This may or may not be true. Can you give me an example? If this guy is true. Then this guy may not be true. One second sir. Two minutes. Two minutes. Figure it out. Oh yeah. So if X died because of corona virus. Q implies P X died. Doesn't mean he died because of corona virus. Okay. I will give you a very related example. If a number is divisible by. Is divisible by. Divisible by nine. Then it must be divisible by three. Yeah. Okay. But can I say the vice versa is true? No. If a number is divisible by three. It may not be divisible by nine. For example six. Okay. So my dear. That time in class nine. You were proving theorems. You had to prove their converse also separately. Are you getting my point? So when I was like studying in my class ninth. All these proofs of triangle and circle. No. There was a proof that if you draw of. You know perpendicular on the chord. It will bisect the chord. Okay. Then there was a converse of it. A perpendicular from the chord must pass through the center. Like that. So I was very frustrated that time. Why are we proving the same thing? You know two times. Because of the fact that if an implication is true. It's converse may not be true. Are you getting my point? We'll soon see the truth table of it also. Okay. Next type of statement which will come across is something which we call inverse. Which is like if not P. Then not Q. This is called the inverse of the implication. Okay. Let me give you an implication. You tell me what is the inverse of it. If it is cold. Then. I will. Wear my jacket. Okay. Tell me what is the inverse of the statement. Tell me what is the converse also. Okay. Devata has given a statement. So the inverse is if it is not cold. Then I will not wear my jacket very good with you. What about converse? If it's if I'm wearing my jacket. That means it's cold. Yes. If I wear my jacket. Then it is cold. Okay. Now the third one is what we call as. If not Q. Then not P. You can call it as the inverse of the converse. Which we call as contra positive statement or contra positive of this implication. Okay. So if I have to write a contra positive for this statement over here. What will I say? If I don't wear my jacket. That means it is not cold. Yeah. Are you getting my point? Yeah. Yes sir. Now just a question of friends. Which of these three means the same thing as the statement itself? Contra positive. Contra positive. Contra positive. Hey who will see who has come? Oh my God. Dheeraj you made my day. Welcome sir. Okay. So the answer to that is yes contra positive. But we will see that truth tables. Let the truth table tell me that this and this are logical equivalent. First of all let us see the truth table of if P then Q. Truth table of if P then Q. So listen to this very very carefully. It may be rising to some of you. It might be. Rising. Surprising okay. If you both are true what do you think is it true or not? True. True. If this is true then it is false. What are the answers? False. False. If this is false this is true. False again. It is true. Oh. If this is false I don't think it will be surprising. It will be surprising. False. True. True. Yeah. Double negative becomes positive. Okay. Doesn't happen the other thing. No. How is it? Yeah. Dheeraj and I will explain you how it happens. Okay. Let's say this husband who makes a promise to you that if I get a promotion. If I get a promotion. Then I'll take you to Singapore for a vacation. Singapore for a vacation. Right. Now tell me when does the husband make a good husband? When he keeps a promise. Of course when he keeps a promise. So let's say he gets a promise. That is true. It takes his wife to Singapore. That is also true. Is he a good husband? Yeah. If it is a good husband write it to there. Okay. Now if he gets a promotion. But he doesn't take his wife to Singapore. Is he a bad husband or a good husband? Bad. Bad means false. Okay. Yes sir. If he doesn't get a promotion. Poor fellow but still takes his wife to Singapore. Nice guy. Super god husband right? Yeah but also poor husband. That fellow. So again write it true. Okay. And if he doesn't get a promotion. And he doesn't take his wife to Singapore. That doesn't make him bad actually. He actually fulfilled what he said. That's true. Okay. That's a funny way of remembering it. But the idea is if the antecedent becomes false. We do not care what is my consequent. Whether it's true or false. We do not care. Your implication will still be true. For example if I say if you study you will clear. Then you will clear IIT. Okay. Then you will get into IIT. Okay. Now if you studied and you got into IIT. Peace. There is no problem right? Peace. But if you studied and didn't get into IIT. Then definitely you will come and shout at me. If you said you didn't get into IIT. I studied but I still didn't get into IIT. So if the antecedent is true but the consequent fails. Then only you will come and shout at me. Then only you will raise a question. Yes sir. But if you study and let's say you cleared IIT. That means you will be still happy right? Because you know antecedent here is false. That doesn't mean your consequent cannot happen. Just like your school. Your school is closed. It may not be a Sunday. Oh okay. Okay. And if antecedent is failing again and consequent is failing. Then it is basically doing whatever is required from the statement. So if you don't study you will not get into IIT. Yeah. Now I would like you both. No not both. All of you here. I actually thought two people are funny so two people are serious in the class. I'm sorry. So now I want you all to make a truth table off. If not Q. Then not P. Please make a truth table for this. And very fine with it. Except for what we got over here. Sir your voice is going in and out sir. Like amplitude is decreasing and decreasing. Really? Yes sir. I want you to give me the truth table for this. When you are ready let me know. Let's do one thing. Why to do only this? We'll do all the three. We'll do this also. We'll do this also. And we'll do that. So that we can check which of the three that we had discussed. Converse, inverse and contrapost. It's logically equivalent to the implication. Okay sir. Sir will this table just be the bottom to top for the above table? We'll check. I don't remember the exact answer. It depends on how you start and stuff. Okay. Can we discuss now? Wait sir. Q implies P. Can we do it together? Okay sir. So let's do the truth table of Q implies P. So we did the reverse table. Promotion takes his wife. True. Yeah. Didn't get a promotion. Takes his wife. Good husband. Yeah. His promotion doesn't take his wife. Bad husband. Promotion doesn't take wife. Not P implies not Q. So read it like this. False false right? Yeah. False false. True. True. False true? False. False. False true true. False true true. Oh yeah. Correct. False. False. Yeah. So you took that antecedent thing as Q right sir? No no no. I'm talking about this guy. Oh okay. Not P. Not P is false. No. Not Q is also false. So false false. Means not getting promotion. Not taking wife. Good husband right? Oh yeah. Okay. Now false true. Means not getting promotion. Still taking his wife. Good husband. True. True false. Getting a promotion. Not taking the wife. False. Bad husband. Getting a promotion. Not taking the wife. Yeah. This one. This one you have to read the other way around. False false. Good guy. True. Top and bottom will always be like same. True. False. False. Yeah. False. False. True. True. False. True. Yeah. Good guy. Okay. Now you can see here very very clearly as TFTT and here also TFTT. While the others are not giving TFTT. What does it mean? Yes sir. It means to us that P implies Q is logical equivalent. So many books write it like this. Many books write it like this. Both are fine. It's logical equivalent if not Q then not C. Okay. Yes sir. Fine. Sir sir I have a request from you but in this page only two minutes you can use pencil. Why? Since you are giving this notes. But one second. So for this. This and this. Like I don't know how to tell this but I won't. For me in future reference replace P with Q. That statement. Yeah. This is your husband's promise and then this is your wife sticking to Singapore. For this guy. Yeah. Okay sir. One second. For my future reference only. I hope no one will mind. Anyway you are giving this to us. Then. Then. Yeah. Okay. One second. One second. Okay. Done sir. That's an E by the way. Yes sir. Thank you sir. What about the negation of this? You write it is not going to be safe. Oh yeah. I realize that. DM. Anyway it's okay. Okay. Negation. Negation of this. I claim that this is the same as saying. Be happening and do not happen. Let's keep like this. Bye. How sir? How? Let's verify this. Okay. Okay. P. P implies Q. Negation implies Q. P and not Q. So let's say true. What will be this by the way? True. True. Good husband. What is this? False. False. What is true? False. What is true and not Q? Not Q is false right? Yeah. Correct. Yes sir. Yes sir. So far false, false same. Let's see true, false. Bad husband. Yes sir. False. Negation of it is true. True. True and negation of false. True and true that is true. True. Okay. Now false true. Okay. Nice guy. This one. False and false. False and false. Yeah. Then finally. False and false. False and false will be true. True. This will be true. True. False. True. False. False. Do you see that there is this table and this table exactly this column and this column exactly match. That means. Yes. They are logically equivalent. And it's common sense also if you say P, if P then Q, then the Ulta will be P happening and Q not happening. Isn't it? When is the only? Yeah. You say, sir, you said if I'll study, I'll get into IIT. I studied but I did not get into IIT. So that will be the negation of what has happened. Yes or no? Yes sir. Kind of, I don't know. Damn weird. Okay, sir. Next is. By conditional. This is basically if and only if. Okay. Now if and only if statements are basically represented as two direction of the action. Okay. It is to be read as P if and only if Q. Okay. That means P is both the sufficient as well as the necessary condition for Q and vice versa. Okay. So we can also read it as P is unnecessary and sufficient condition for Q and vice versa. Okay. You can say that this statement is as good as if P then Q. And if Q then P logically come up and operator or logically come up with a conjunction. Yes sir. So let me give you an example of this. If you give bank cut. Oh yes. Fruits. If and only if. If and only. By the way, this is represented by short form with IFF. Okay. Yeah. And bank cut will eat. So that means bank cut won't take it from tree and eat. I should give it to him. Yeah. So now. Now what is the meaning of this? I think I wrote your spelling wrong. Now what does it mean? It means that if you if bank cut. Then only he will eat. And if bank cut has eaten, he must have eaten fruits. That means both imply each other. So if somebody comes and tells me, hey, then I would understand automatically that when cut was offered fruits, then only you would have eaten as not correct. Okay. Yes sir. When it was given fruits, then only you would have eaten fruits. Okay. Then I would automatically that when cut was offered fruits, then only would have eaten as not correct. Oh okay. Yes. He would have given beans. If and only if he given cut, ah, yeah, yeah. Oh okay. I thought I thought the main thing was give here. It's not give it's if and only if does the things. Yeah. It doesn't mean that the, he'll go and pluck from tree That doesn't mean that right? Don't go into the lateral part of it So if Venkat is given fruit He will eat Think like that Oh yes sir Don't focus on the give part Yeah Offered fruit or if he is provided with the fruit Then only he will eat Yes sir Think like that Okay Can we have a tooth stable of this quickly? Yes Let's have a tooth stable of If and only if P then Q Remember that this is equivalent to saying The conjunction of this P, Q and QP Yes so keep this in mind while you are making the tooth stable If you want I can make Separate columns for these two And then we can have a column for If and only if Let's do this True True What is this? True True True This is also true But if and only if it is true Don't both have to be true or both have to be false I will come to that That's what we are going to discuss now So if this is true this is false First of all bad husband Yeah Good guy Intersection or you can say The conjunction of these two will be false False Okay False to good husband Other way around it will be a bad husband And conjunction of this will be bad False False And finally when you have false False This will be true This will also be true And conjunction of true True is also true What does it mean? It means as what did I rightly pointed out Unless until both the component statements are true Or both the component statements are false It will never be true Okay So this is like what This is These are the kills of or statement Right And this is separate class These are all the subsets of or statements Where is an or statement? This is an and statement This is an and only no No no Okay Because in or false and false gave up In and false and false gave up false Here everywhere else false and false giving true No for and statement False false was also a false That's what yes Or was also false false false false But here it is giving true Okay sir So basically when I say If Venkat is given a fruit Then he will eat Logically if you were into saying If Venkat is not given a fruit He will not eat Are you getting my point? Yeah Okay Yeah Yes sir No If I ask you Tell me the negation of this How would you write? Okay What do you think should be the negation of this? Don't worry too much The negation of this is The negation of this Yeah the negation of this guy Right Oh okay You know when two statements are connected By a conjunction You normally write it as Negation of Disjunction Yes sir And this itself is P and not Q Okay Or Q and not Not P So do it for homework Brow the truth table for this And check whether the truth table for this Gives you false to true false Hmm Okay sir One second I will take screen shot of homework Please Done sir Now Somebody was asking What's the tautology? Tautology is basically a Statement which is always True What is the truth value What is the value of the Confident statements Should I repeat my statement once again? Yes sir A tautology is the statement Which is always Irrespective Of what is the Truth value or the Value of the component statements Irrespective of what is the truth Value of the component statements Okay Let me give you an example Let's say P Or not P Is it a tautology? So here P Is the component statement Okay So Let us start with table for this So this can only take true or false Right So the negation of it will be For or true And if you take the or it will be True or true Irrespective of whether this is true or false This guy is always true So this is actually Tautology We will definitely ask you Okay I am just giving you an example On the other hand There is something called Policy Or we can say contradiction That will be that guy Deplace a downward arrow with upward arrow Right An example of it can be P And not P That means the statement will always be false Irrespective of whatever is the Truth value of the component statements So let's say I take this as True And a false then this will be False and a true and there Will always be false False false Whatever is your input statement Your output is always false This is our example of a policy Yes sir Now before we proceed I will take up a lot of questions With you And then I will start with the Questions Let's start Questions Let me just open to the sheet here Can we quickly complete The archives Previous year papers Yes sir First one So Transdental is at Pi in all right What's a Transdental number Let me think sir Transdental function Or Stand function Sometimes in back Transdental Something like e to the power Sir is Pi a Transdental thing Sorry Is Pi a Transdental The actual Pi Not the 22 by 7 Pi See basically Yeah Pi and e are Transdental numbers I mean the actual Pi right Yeah yeah yeah See basically in mathematics we define Transdental numbers as those numbers Which are real or complex That is not a real problem Non-zero polynomial equation With integer coefficients Which is real or complex Which is not a non-zero solution Of polynomial with integer coefficients Integer coefficients But is that required here For problems or like No sir Read the statement Yes sir Pi is a statement where it says X is an irrational number Q is a statement Where Y is a Transdental number And R be a statement Where X is a rational number If and only if Y is a Transdental number Okay Now these are the two statements That they have made R is equivalent To either Pi or Q No Basically that means my answer is A Yes sir Yeah Let's check it out So if you make two tables of it R is an irrational number Okay that's what Pi says Correct But this guy says Not Pi Implies Q Correct Yeah Question is saying That these Two statements are equivalent Oh By the way Q or P or P or Q are same thing Because they are Yes sir Can you make a two table of this And check it out Yes sir Here itself Not Pi implies Q And here I will make Pi or Q True true This guy will be True first one will be true And this guy will be a True false True false False and Good guy True false is going to be true here Yeah good guy This is Not P and Q right False false Now false true True Correct Yes sir False Now take the order of these two True or true is true True false or true is true False false is false This is Oh yes sir That means statement one is true right here Yeah Your answer is wrong Yeah What about R is equivalent to Saying Not Not of or negation of If and only if P Then not Q That means you are saying This is same as This guy Check it out Yes sir We will erase something over here Necessary things Okay do you want me to make this again It is already made right Yes sir And I will make Negation of Okay now in my mind I will do lot of things Please feel free to ask me So first you see P If and only if P then not Q So that tinda P Then that double arrow Q In your mind treat this as false So true and false Will definitely give you a false Right Until both are true or both are false This will not give you a true Yes sir This will give me a true Tinda of that yeah Now true true will give me a true This will give me a false Moment I see that there was a true here And there was a false coming over here So this statement is false That means option D is the right option Yes sir Others who are silent Can I take your silence for That you have understood it Sir I Sorry sorry sorry Vidyota Yes sir Type yes personally to me Yes sir Everybody should type yes Is it okay if I say yes Yeah I am only able to chat only How long has it gone I think Yeah sir the chats are only private to you Yeah I know Oh you kept it like that Thank you sir P implies Two implies P Equivalent to which of them Moment equivalent comes Please make truth table Without truth table do not Take any kind of a guess Okay sir let's do the truth Sir I am not sure I understood the question though Like how do you Identify the roots of the statements and all Or is it rational if and only if Why is it transcendent number So let's P Double ROQ I will take the snapshot of this And pull it to the Why screen Now the best way to solve such kind of question is You should make a truth table with all of them P implies P implies P implies P implies P or Q P implies P implies P implies If and only if Q If and only if Q Okay don't be scared Just driven by truth table Okay don't be scared Truth to Let's feel very slow okay First of all tell me what about this Q implies Q Yeah Truth Now P is true and this is true Husband getting a promotion taking his wife Good husband Yeah yes sir This can also be said to be true Yeah What about this This is true this is true Yeah everything is true in this thing True true true true Everything is true We have not been able to distinguish between the options Basically Yes sir In true false In true false What about this What about this What about this One second sir He took her to sing No he didn't take her to sing No he didn't take her to sing No he didn't take her to sing No he didn't take her to sing No he didn't get a promotion Still took his wife He is a bad husband Read Oh correct Okay yeah Yes sir Good husband So true True true Is true True Now this will be false True Getting a promotion not taking his wife This will be false Yes sir What about this P or Q will be true True true will be true Correct Am I going too fast No sir But Yeah okay yes I got it We will go back This was clear right Yes sir Okay now this is true this is false So what should be the answer for this Are they say something Oh sir I muted myself True P implies Q when P is true And Q is false Nehati already answered it False Now this is true Now think as if this is the husband's promise And this is the taking of the wife to singapur The husband's promise not taking the wife Okay bad False Yeah yes sir Now do the same thing over here What is P or Q True True And this is already true P is already Getting a promotion Correct The moment you realize that this is not Mapping with this that means this cannot be true Option A is ruled out So Going forward I am not going to waste time On this option This is crap Let's see what else we can Now this is going to be true And this is going to be false right Yes sir This is also going to be Correct no Yeah This is true This is also going to be false So this is also going to be scrapped Yes sir You don't have to do any further Don't waste your time Because I will not end up giving you anything You just have to mark option B And go on Yes sir Make sense to you or not Everybody So it makes sense It's just a little slow Takes a bit of time to write Yeah takes time Another ways yeah Third one Statement one says negation of If and only if p then q is equivalent to If and only if p then q Okay If and only if what negation of If and only if p then not q Is equivalent to if and only if p then q That's false That's false How do you can do this so fast Sir Then you Try expanding this That thing p That double arrow Double double q Okay If this is wrong then you hide it from me You said First statement is false right Yeah If this is wrong Then Then see what I do to you Okay sir That means it's going to be false My statement No it may be correct also But you should not jump so early Yes sir It could be correct also I'm just kidding you Sir I will not I'll do some calculations In my mind also Let's take So this is false Okay now what about this Both yeah Two false it has to be false because this is Both are true or both are false Yes sir This is true And this guy will also be true Okay Up to true True false True True And now see here true true True This guy is false And now look here True false Into you False This is false False Now look at this guy and this guy False false means true Negation Now these two will be false And that False false This is true Then this and this will be false False That will be true False false is true True See don't jump into conclusion So this is a true statement Yes they are equivalent Okay So statement one is true True Now do the statement number true Okay yes sir Is that a tautology Hey you Tautology Tautology Okay one second sir PQ It means whatever you put your answer for this will always be true. Is that the case? Check money So let's say True No sir Tautology actually are correct No See right See is the answer Next if alpha beta are the roots of this This is not a This will do This seems like a Minus omega and minus omega Minus omega square yes Just put it in You have already done Okay so we will do 6-1 6-1 6-1 is Consider the following statement Suman is brilliant Suman is rich Suman is honest Which of the following statement represents Negation of the statement Suman is brilliant and dishonest If and only if Suman is It's very obvious See from the options Disonest Pima Pima What do you think I am thinking It's I am thinking If and only if Suman is rich Okay A sir Suman is brilliant and dishonest Sir total the answer one second Brilliant is dishonest P P that and Dash R If and only if it is rich So basically this by condition with If and only if it is rich Rich is Q right This is your answer Yeah Not this one Not this one Yeah this one see Oh sir why can't we solve it like P of Thing Dash R maps to Double maps to Q Conjunction not R Sir Now it has conjunction Conjunction yeah this guy is conjunction See here Suman is brilliant and dishonest This is one statement which is connected With if and only if Suman is rich Correct So Q Suman is rich If and only if symbol Suman is Brilliant is P P and Dash R okay Disonest is negation of R The whole Things negation you want So this is the answer Oh yeah Yes sir Next The negation of this Then I will open a school Hmm I told you that negation of P implies Q Same as saying And not Q And not Q yeah And I will not open a school Open a school yeah Yeah See what type of questions were coming in the past So easy very scoring topic Not leave out any Sir the subjective questions were hard And then from 2000 to 2000 Like what 11 it became damn easy And now it became harder again Oh really Like 1900s 1970s Now I just seen one question That's they were hard And 2000s to 2011 As long as AI tripled it became damn easy Then again it started becoming difficult Oh is it like that Yeah Second analysis I saw some questions sir Next Consider statement one And statement one is a Falsy statement two is a totology Statement one is true Sorry sir I don't think I was listening right What's the totology again Oh my god Totology is something which is always true Falsy which is always false Okay got it thank you sir Outcome of this if you do a table for this It should always give you false false false Okay Yeah that seems true one second sir That was an honest confession Just let me know when you are done with the first one Statement two is also true Statement one First one is true Both statement one and two are true Both statement one and two are true Both statement one and two are true Not sure about the explanation Okay one second bro I'm still doing the first one You know one and two are both are true This one I know According to you the B should be the answer right Correct explanation There's always confusion Yes sir but once again I won't I'm not yet given my verdict on the first statement Okay I'm waiting for your verdict Supreme Court I'm not False false What is it again True true true True true true True false false False false false false False false false false Yes sir First one is false I mean it's a fallacy is true Okay shall I will check So P and not true This is false FT FF This is going to be T This is going to be F Not Q is false True is false Okay so I'm slightly faster Then you're taking the conjunction of this So this will be false False false false This is definitely a False This is our true statement Okay Next one Total G So the sad part is only one option gets eliminated in these kind of things Okay we don't have to do this also This is a Contra positive of the same statement Okay I didn't see that Let's do the Let me call this statement R and this as This is R itself right Yes sir R actually Okay this will always be a true right R implies R Yeah Let's say this gets true Then this will also be true This is false this will also be false Yeah The true true will always be true False false false This is a tautology But they are nowhere So you can see So absolutely Direct Sir You're absolutely correct Statement 2 is true But 2 is not a correct explanation for one Ninth one Let's do ninth one Okay sir Uncle G T B Bad buddy I'm not that fast bro You guys said B right No no no sir one second Others also please type in your response if you don't want to speak up This is just a second One second sir That just became an F That became Yeah I'm gonna say I'm gonna say A Before I say anything I won't say anything Now keep your options ready with you One second sir Before one second please T F Yeah I guess I'll say B But I need more time which is sad Sir I'm redoing it sir one second I messed up somewhere F T F Okay What about the present one Have you done this present one T1 second almost Where does this guy go Yeah so this guy This guy Sir I'm saying A sir Yeah I'll say A I'll say A sir Can we discuss quickly Yeah so this will be P This is false F True true will be False false will be true This will be also False Negation will be true False I think there was True false false true This is equivalent to A Yes sir Ideally I should have made all of them But the first option I just Tied it out and it was matching A is the correct option Sir totally false I can Remove them Yeah This is only one thing right So one time it will be a false But not take the risk but sure Last one Negation of This guy Is equivalent to Sir I'm pretty sure no one like the Sir That's why they were wondering Okay with Dutta I noted down your answer Dutta has already answered I just finished Writing the titles of the Truth paper Things you can directly apply You can apply that what do you call Murugan sir Yes And negation of this Negation of this will be R Oh That That is here This is like your Think as if it is S intersection R union S complement S intersection R Union S intersection S complement But this is the null set Null set Answer will be this You don't have to actually Option D Thanks to the property Of sets that we can The twice of the symbol was also Because it was resembling your union And intersection this is like an intersection This is embers like that this is like your union Yeah Okay Anyways last part is left we'll complete That and we'll do more questions Don't worry after we have completed this We'll take up more questions The last part of this topic is validation Of statements Validating statements This is important from your school point of view But it will not be asked in J Actually Because this is Proving it by writing Statements and sentences Which definitely will not be required In J Okay Now How to validate two types of statement What is your if then implication And the second one is your if and only if Statement What is the meaning of validation Validating Proving it is true First let us start with If then So if you have been given any statement Which is an if then implication That is if P then Q Then we normally have Three ways to prove it Okay The first method is what we call as the direct method In this method what we do We follow these two steps We first assume That P is true Okay And then By assumption in the first We prove that Q is true So if my both P and Q become true-true We know that if P Then Q has to be true So it works on this particular truth table Let me give an example Let me give an example on this Let's say Let me pull out A good example For this N is an even number Then N squared is an even number I can say even integer Okay Now let us say we want to use our Direct method To validate this Then statement So When your train is cool Always first mention what are you calling as P and what are you calling as Q What is P here P is because N is a N is an even number Even integer And what is your consequent N squared is an even integer Okay Keep it as simple As possible Now By direct method We will first say let P be true That is N is an even number N is an even integer So let N be 2 Let's say M Now what is N squared N squared can be written as 4M squared Yeah So now we can write this as 2K Where you are calling K as 2M squared Which definitely implies that N squared is even That means you have proved that Q is true So by assuming P is true If you have been able to prove that Q is true That means it implies This is also true By this straight truth table Okay Now many people ask P was actually not a true statement Now let us say it was false also But if you prove that your consequent is true Then still false True will give you a true Remember Still took his wife for the Singapore trip That's still a true statement Okay So here assumption itself being false Doesn't matter to us Are you getting my point So this implies This is true That means this statement here is a true statement Understood how direct method works Now The next method which is B So A we have already discussed Direct method so I will go up Another method which we call as the Contra positive method Contra positive method All the steps that we follow for Contra positive method In Contra positive method Step number one Let Q be false That is you are saying Negation Q be true Okay And using this assumption By your first assumption If you prove that P is false That is you are able to prove that Negation P is true Then basically what you have done You have done something which you did In the direct method but now applied To negation P and negation P So that means this is true And since this is logically Equivalent to this It means this is also true Hands done Now you must be wondering One second sir Let Q be false Okay Prove that By using one prove that By using the first assumption Prove that P is false Negation P is true Yes sir That means you are proving that If not Q then not P That is not possible Of this And both of them These two statements Are logically Equivalent to each other Yes sir Therefore If this is true this guy also has to be true Because we follow the same proof tables Correct? Now where do we need A counter-positive approach Let me go back And give you the same question In a slightly different way Listen to the statement R which says If N square is an even number Even integer Then N is also even Okay Now try to prove this If you use direct method You will see that there are some Bottom lengths For example let Let N square be Even I can say let Let's say this is your P And let's say this is your Q Let P be true If you say that you are then saying N square is equal to even number Let's say it's 2M Then you enter Under root 2M Now we are clueless How do we show that this is also even So direct method It is not a convincing way Of doing it Because you will not be able to Convince the example that If 2M under root is also Under root Isn't it? Yes sir So this has a Shortcoming But we cannot use direct method So let us use counter possible So in counter possible Let Q be false Okay That is N is not even number But N square is even Wait So let N be odd By the law of X2 If something is not even it has to be odd Correct So let N be 2M Correct So N square will be 2M plus 1 square That's nothing but 4M square plus 4M plus 1 That's nothing twice of 2M square plus 2M plus 1 Even always 2K plus 1 That means it is odd Correct Yes sir Which implies you have proved that It is false Correct That is Proof that negation is True Correct So by taking an assumption that Negation Q is true You have proved that negation Oh So we can say that This is true And hence This is true Because both are equivalents Are you getting my point how it was Yes sir Okay The last method which is not a new method to you You have done this in your childhood days also Which is called the contradiction method Aiyo Do you remember that Real number chapter you had in series In class 10? True that root 2 is irrational So how does it work in this case When you are proving A statement if Then statement to be true By using contradiction method The approach that we use is We first assume that P is false and Q is true What do we assume P is false and Q is true Or we can say P is true No And Q is false Why have I chosen this Actually In any Your sense See This was the only condition when your written Statement was false right Remember the truth Husband gets a promotion Doesn't take his life for the foreign trip For the Singapore trip This was the only situation when This was false isn't it Yes sir I have assumed that let Q be false That means that we have assumed That this is false Okay And then I reach a contradiction Step Or you can say contradictory step That means this is not true That means This is happening That means negation of false is happening Let me write it false That means your statement will become True Are you getting my point So again I am repeating the process First you assume that P is true and Q is false By assuming that you are Indirectly assuming that you are given If then statement is false And when you contradict that Means your if then statement becomes True simple as that And hence we prove that this is true Yes or no Yes sir A Don't give such a cold response What is Not clear in this With it Bandwidth is not matching One second If P is true and Q is false Yeah Got it sir Can you do the same question Which I had given Let's start with this guy Okay let me give you another question On this Let's see whether you are able to do that Yeah Question is If XQ plus 4X is 0 Where X is your real number Then X is 0 Validate this By All the three methods Direct method Contrapositive method And contradiction method But first start with contradiction So that we have understood how that method works Sir is there a method called solving method No no Okay sir Then you come to XQ plus 4 which is That complex thing but then you are given the condition Then we will discuss it Yes sir I have done it not sure if it is right or not Yes sir can we discuss no sir Yes sir I did something I am not sure Okay we will start with contradiction method Because we haven't done this yet So in contradiction method What were the steps? So first we will assume P So this is your P by the way And this is your Q False It is better to write them very very clearly So P and this is your So in contradiction method Let P be true But Q be false And Q be false And yeah correct That means you are saying This guy is 0 And this guy is not 0 Right Yes sir That means you are saying X into X X square plus 4 equal to 0 Since this guy is not equal to 0 This guy should be equal to 0 Now we know that Now we know that If X not equal to 0 Real number X square plus 4 Will always be a positive quantity Isn't it? Yes sir That means if the product of X with a positive quantity is 0 That means only X could be 0 And this Since X is not 0 Isn't this a contradiction At one place you are saying X is 0 And at other place you are saying X is not 0 Yes sir The assumption is It's self-false That is That implies Q is True Because the assumption is That implies Q is false And you have proved That false itself is false Means it is true Negation of a negation False itself is false Basically what you are saying You are saying negation of a false Yeah Not false means it is true Yes sir Now Can we discuss direct method And counter positive method as well Are you done with that or do you want some time Some time sir Do direct method now This is what I meant by saying Solving method Basic solving you have to do Yes sir So counter positive done What are you doing No sir Do it anyway Okay Let's try direct now I mean I didn't write anything for counter positive I just thought of it Guys are we ready to do it with Direct method Sir could you just close up with the direct method page one second Okay direct method is assume p is true And then prove Q is also true Done sir We will discuss direct method now In direct method what do we say Let p be true That means Let xq plus 4x be equal to 0 Again same thing That means you are saying this is equal to 0 But since this is always positive This always these two x equal to 0 That means p is That means q is true So by assuming p is true We have proved that q is true Therefore This If p then q is also true Done simplest Next is the counter positive method So what do we do in this case we will say Let q be false Yeah let x not equal to 0 That means you are saying Not q be true That means you are saying x is not equal to 0 Okay So now That means x is either positive or negative Now This can be written as this Okay So if x is positive Then x into x square plus 4 will also be positive Because this guy is always positive Okay And if x is negative Then x into x square plus 4 will always be negative Either of them implies that This term will not be 0 Yes sir That means you have proved that p is false That is you have proved negation p is true Right And hence we can say negation Negation q implies negation p is true Which further implies That this is true Because they are both equivalent statements By the counter positive approach Is that fine Yes sir Now we will do some problem solving On the type of questions That we will be getting in the exam Yes sir Go to the First few questions D First one Are we ready with the answer Yes sir Which New Delhi is the city which is calling D D for Delhi All of them are negation of p Yes sir Because they are asking not a negation of p Yes sir Is that fine Yes sir If p is a statement and which of the following Is it tautology Is it tautology Is it a policy One second sir F means A statement which is false This is One second C None of these Is this a tautology No sir Is this a tautology No If f is false then false or false Which is bad If p is false sorry Sir It depends upon what is p If p is false it will be false The entire thing So it depends upon p here So this also often d will be correct Yes sir Okay Let's skip the third one and we will move on to the next Third one is c by the way Correct Which of the following statement Is true If each of the following statements is true then What have they written here Some mistake is there That seems to be right Wait sir I don't know p means not q Okay I didn't even get that So Just need these two I think some printing mistake Let's go to the sixth one Equal means tautology p and true and false This is a tautology sir C sir no sir no sir what about this P no sir tautology C Q P that's Q is equal no no no not D sorry D is correct yes sir this is the this is this is the best. Contrapositive. Contrapositive. Next, 7th one. P and Q are false P and Q are false intersection cannot be true so this is wrong. Union can be false. False right? Yes sir. Is this correct? Hmm. Okay sir multi-choice or single-choice? Let's see it may be multi also never know. J advance as this chapter. Okay okay. Yes sir. Yeah. Damn bro. Both P no no no. Gone. Okay same thing. We can do 9. P is 2 Q is false. Which of the following is not true. That means false. That means false. Yeah. P or A and B. This is false. Hmm. False. That one no that one no. This is basically the answer this guy so this cannot be the answer. This is a tautology right? Yeah. This cannot be false. So A and B are correct. Yes sir. It's complete this exercise. Okay sir. Which one you want? Converse. What's that? 10th we did. 10th we did. P is false Q is true. This is true is wrong. No. This is false. Yeah. This. No no no. This. Hmm. Yeah. It must be. So 10 D. Next. P implies Q is true. How sir? How P implies Q is true? Husband didn't get a promotion. Okay. He took his wife for the foreign trip. Yes sir. Yeah. Next. Converse of this statement. P or this one. Okay. What is that? Converse. Sir finding contra positive is similar to reflection about origin as is realized. Widyotha. In two steps. What is the answer for the 11th one? Widyotha and Advaita. Converse of this. While I was explaining I told you know what is that? Converse is Ulta. This is called. Yes sir. This is called contra positive. Contra positive. It's called inverse. Hmm. Yes sir. Next. Contra positive. Widyotha and Advaita both please write down or you can speak up also. In fact let. When that you also write down. Nobody will get in this place. Okay sir. 12th one. Right? 12th one. Yeah. By the way when you are writing write as the person number and your answer. So when I know it from your answering. Okay sir. Next. Which of the following is logical? Negation or negation. One second sir. Okay Widyotha. Okay Widyotha. Okay down. Let's wait for others. One second. Advaita what about you? One second please. Oh sorry sorry sorry. I shouldn't have said that. Discuss correct? Let's discuss. So negation of negation implies. Okay. So true true. So false true is a good husband. So this is false. Then true false. Then false. Then you have false true. That means it's true true and this will be false and false false. Okay. Now this is the truth statement of which of the following. Which of the following. True true false. Doesn't give me. Yes sir. So if A is not the answer why will D be the answer. A is not the answer why will D be the answer. Contra positive right? Contra positive is not Q implies not P. Yes sir. Yes sir. This is the answer in between. Okay. Now not P means false true. False true will give you a false. So this is matching. True true should give you a true but there's a false over here. So C also cannot be the option. Now false and false is false that is matching. True and false is false and true that is matching. True and false false again matching this is matching. So D is the option actually. Do you want me to do it separately in some different page or is it understood from here. Understood sir. Next. Which of the following is true for any two statements P and Q. This is a symbol for equivalence. This is a symbol. The same that we use for congruency. Okay Vidyota. Multiple may be correct. So please be careful. Okay. Okay sir. One second. This is not all of it. Second question is incorrect. Yeah. One second sir. One second. Almost gone. Sir a contradiction here is what? I understood what contradiction actually means but what is it in this context here? In this output will always be false. Oh okay. That means policy. Yeah. Okay sir. One second. Yes sir. This is incorrect. Yes sir. A and C. Okay. Gone. Okay. Let me check about the C option. Yes sir. True true. This will be true but this will be true and false will be false. Okay. Next is true false. This will be false and false. false and truly false. false and truly false. And this will be false. The false and false will be false. And finally you have false false. So this will be false but this will be true. So this is a contradiction C is also correct. Yes sir. So Vidyota why did you leave out that symbol that they have made? Let it be the symbol. Sir are you able to hear someone crying? Yeah. Okay then I should mute myself. Baby actually. Okay Vidyota what about others? One sec sir. I just now figured out how to do it. Okay let's discuss. Yes sir. Substitute options right? Yes you can do that only. So let's say true false false and C is true, Q is false and R is true. R is also false. True and true. Yes sir. And this is and this is false and false which is false. The true and false. False. Okay. What are these? False and true gives false. No no no PQR false and false is false again. Oh yeah. And true and false. True and false is false. So false and false is false. So this is also correct. Next. Next. False and false. So this will be false. Yes sir. This will be true and true. So false and true will be false. False again. This I don't think. So this will be true and false. This is false. The moment one is false. That is also false. Yeah. Here we could have saved time by checking if any one of them is false. Oh yeah. Next. Total. 16th one. So can something be a totality and contradiction at the same time? No rate. Correct. Yes sir. No rate. That's the whole genesis of calling a statement because we're dealing with a statement. A statement cannot be simultaneously true and false. Yeah. Just a second sir please. Done. So much time. Yes sir. I sent you. That's correct. 17th. Correct Advaita. It has to be a totality because both are same things right. Yes sir. So 17th is the right answer. Correct. 18th. Sir in the third option 18 should we assume some sign between there. Oh yeah they are missing out on the sign. Which book sir. This is the Arian. Cengage. Arian. Cengage okay. Cengage okay. I don't think this. Yeah. Okay. What is P? P is false right. P is false. So wherever there's a negation of P there should be there. So P is false. 6th is the divisor of 12th and this is also true. SCF of this is true. This is also true right. Which is true. This cannot be true. That can't be true. Sorry. Yeah. Correct sir. Yes sir. Nothing nothing. This is true intersection false. This is really false. This is also true. And this is true. True or true is true. That means option D is your right answer. Yeah correct sir. 19th. I'm sending you sir. Others. Okay let's discuss here. We already know the truth table for this. Hmm. What is that? True true is true. True false is false. False true is true. False false is true right. Hmm. Just remember this. Okay. What about 18th? Other way round. True false true right. If I take all of the same first. Hmm. The true and true is true. Okay. Hmm. This will be true. Then all and true will be false. Yes sir. Then true and true. This is not logical. Yes sir. Let's take this one. P and Q will be true or okay let me write it like this. True true or will be true. Good. Now if you have true false for both of them this will be false while this will be true. Hmm. And then or will be true again which doesn't match with this. This also cannot be wrong. Yes sir. Last one. Sorry B1. P or Q will be what? P or Q will be true. And this. This is a tautology right. So this also cannot be wrong. Oh correct correct. This is a tautology right. Yes sir. Okay. Didn't see that option earlier. Okay. What about this one? Here you are taking the union option. So true true is true. False true. False true is true right. False true. He didn't take her but he got promotion. If this is true and this is false that means I am taking into account this case. Hmm. True and this is false correct. Yeah. Then your answer is going to be true correct. This is the other one. He didn't get but he took. If this guy is false and this guy is true my answer should be true again but here the answer is false. Yeah. None of these are logically given. Some again mistake they have been. Yes sir. Yeah. So can you just check the A1 again. I feel that might be the wrong answer. Okay I will check again. I think it is the wrong answer. I will take this question separately. Okay. Yes sir. So let's have P. Q. P implies Q. Q implies P. And then you are saying to check A right. So P implies Q. Or Q implies P. So if the truth stable of this matches with this that means statement A is correct. Let's check. True true true. False false true false false false true true false false. Sorry true false true yeah. If you look at Ulta, then this will be true, true, false, true. Now they are taking their ORs. So OR will be true, true, true. This is actually a pathology. Okay sir. I mean this is not matched. No not yet. Yes sir. If you take their ANDs, you will realize true, true, false, true. Yeah. It also doesn't match with this. Oh they should have given q equal to implies p and then probably and here and then it will be correct. But two mistakes didn't make it right this time. Let's do 21.