 This is basically if and only if statement are basically represented as two direction of the action. It is to be read as p if and only if cube. That means p is both the sufficient as well as the necessary condition for q and vice versa. So we can also read it as p is a necessary and sufficient condition for q and vice versa. You can say that the statement is as good as if p then q and if q then p logically come together and operate or logically come together with a conjunction, right? Yes sir. So let me give you an example on this. If you give Venkat fruits, if and only if, if and only, by the way this is represented by short form with IFF, then Venkat will eat. So that means Venkat 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 Venkat, then only he will eat and if Venkat has eaten, he must have eaten fruits. That means both imply each other. So if somebody comes and tells me, hey, sir Venkat has eaten, then I would understand automatically that Venkat was offered fruits, then only he would have eaten, that's not, correct? Oh, okay. Yes sir. That Venkat was given fruits and I would automatically understand, yes, he would have eaten. It's not only if he'll give Venkat, oh, yeah, yeah, oh, okay, I thought, I thought the main thing was give here, it's not give, it's if and only if, that's the thing, right? Yeah, yeah. It doesn't mean that he'll go and pluck from tree and eat, 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. If Venkat is offered fruit or if he is provided with the fruit, then only he'll eat. Yes sir. Think like that. Okay. Can we have a tooth stable of this quickly? Yeah. Let's have a tooth stable off, 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're making the tooth table. 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. What is this? True. True. What is this? This is also true. But if and only if it's true, don't both have to be true or both have to be false. I, it will come to that, Veeraj. That's what we're going to discuss now. So if this is true, this is false. That's what we're going to discuss now. So if this is true, this is false. First of all, bad husband. Yeah. Good guy. Yeah. Intersection or you can say the conjunction of these two will be false. False. Okay. False too. 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 is also true. What does it mean? It means as what Veeraj 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 kids of our statement, right? And it's a separate class. These are all the subsets of our statements. Where is an or statement? There's an answer. No, no, no. Okay. Because in or, false and false gave up because 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. Yeah. That's what you said. Or was also false, false, false, false. But here it is giving true. Okay. 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 negation of this? Don't worry too much. The negation of this is the negate. 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 p. So do it for homework, draw the truth table for this and check whether the truth table for this gives you false to true false. Hmm. Okay sir. Okay. One second. I'll take screenshot of homework sir. Please. Done sir. No. Somebody was asking what's the tautology? Tautology is basically a statement which is always true irrespective of what is the truth value of the component statements. Should I repeat my statement once again? Yes sir. A tautology is a statement which is always irrespective of what is the truth value or the logic 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 a component statement. Okay. So let us draw a truth table for this. We can only take true or false, right? Correct. So the negation of it will be false or true. And if you take the or it will be true or true, right? So irrespective of whether this is true or false this guy is always true. So this is actually. Yeah. Now don't expect such easily. We'll definitely ask you. Okay. Let's give you an example. Policy or we can say contradiction. That'll be that guy. Replace a downward arrow with upward arrow. Right. An example of it can be p and not p. That is the statement will always be false irrespective of whatever is the truth value of the component statements. Let's say I take this as true and a false. And this will be false and a true and then conjunction will always be false. False false. Whatever is your input statement, your output is always false. So 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'll start with the validation of. Okay. Let's start. Let me just open to the sheet here. Can we quickly complete the archives? Yes. First one. So trans dental is that by an all right. What's a trans dental number? Let me think sir trans dental function or stand function. That's what we learned sometime back. Something like e to the power. Okay. Sir is pie a transcendental thing. Sorry. Is pie a transcendental. The actual pie not the 22 by 7 pie. See basically. Yeah. I mean the actual pie right. Yeah, yeah, yeah. See basically in mathematics we define transcendental numbers as those numbers which are real or complex. That is not a 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 only. No sir. Read the statement. Yes sir. P is a statement where it says X is an irrational number. Q is a statement where Y is a transcendental number and R be a statement where X is a rational number if and only if Y is a transcendental number. Okay. Now these are the two statements that they have made R is equivalent to either P or Q. No. So basically that means my answer is A. Yes sir. Yeah. So let's check it out. So if you make two tables of it. R is an irrational number. That's what P says. But this guy says not P implies Q. Correct. Yeah. Now what is my question saying? Question is saying that these two statements are equivalent. Oh. By the way Q or P or P or Q are the same thing because they are. They are the same thing. Okay. Yes sir. Can you make a two table of this and check it out. Yes sir. I'll make it here itself. Not P implies Q. And here I will make P or Q. True. True. True. This guy will be. True. First one will be true. And this guy will be a. True false. False. Yeah. True false false and good guy. Okay. True false is going to be true here. Yeah. Good guy. Yeah. This is not P and Q right. False false. Yes sir. Now false true. Once again true. Correct. False. Yes sir. False. False. Now take the order of these two. True or true is true. True. True. False or true is true. False false is false. False false. This is P where is this. Oh yes sir. That means statement one is true right here. Yeah. Yeah. Your answer is wrong. Yeah. Yeah. What about 62. R is equivalent to saying. Not. Not off or negation off. If and only if P. Then not Q. That means you're saying. This is same as. This guy. Check. Check. Check. Check. Check. Check. Check. Check. Check. Check. Check. Yes sir. We'll erase something over here. The necessary things. And did you want me to make this again? It is already made right? So this column erase that Tool and I'll make. Negation off. Okay now in my mind I will do a lot of things. Please feel free to ask me. So first you see P. Only if p. Then not view. So that's tela P. Now in your mind treat this as false. So true and false will definitely give you a false, right? Yes, until both are true or both are false, this will not give you a true, correct? 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. False. The moment I see that there was a true here and there was a false coming over here, they are not equivalent. 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 am sorry, sorry, sorry. Vidyota. Yes, sir. Type yes, personally to me. Yes, sir. Everybody should type yes, everybody should type yes.