 9th one. Let's do 9th one. Okay sir. T. 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. Yeah, I'm going to say B. I'm going to say A. But 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. Yeah, T, what about the present one? Have you done this present one? T, one second. 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. Yes, sir. This is false. F. Correct. So true, true will be T. False, false will be true. This will be also... False. Negation will be true. Mirage, yeah. I think it was true, false, false. False, true. So this is equivalent to A. Yes, sir. Ideally I should have made all of them, but first option I just tried it out and it was matching. A is the correct option. Sir, totally for AC, I can remove them. Yeah. This is only one thing, right? So one time it will be a false. I don't want to take the risk, but sure. Last one. Negation of this date. Is equivalent to... Sir, I'm pretty sure no one liked the sofas. That's why they were wandering. Okay, yes. Okay, with Dota, I noted down your answer. Dota 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? Murgan Shah. S and negation of this. Negation of this will be R. Oh, R that dash S here. This is like your, you know, just a little talk of it. So think as if it is S, intersection, R, union, S complement. But this is the Nelset. Universe. Nelset. Answer will be this. You don't have to actually bother with it. Option D. Thanks to the property of sets that were there. The choice 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. Last part is left will complete that and we'll do more questions. 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 G actually. Because this is writing, you know, statements and sentences which definitely will not be required in J. Now, we'll be learning how to validate two types of statement. One is your if then implication and the second one is your if and only if statement. What is the meaning of validation or validating? It is true. First, let us start with if then. 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. 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 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 we pull out a good example for this. If 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 if then statement. So when you are in school always first mention what are you calling as p and what are you calling as q. What is p then? n is an even number. Even integer. You can write integer. Okay. 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 we are saying n is an even number. n is an even integer. So let n be 2 let's say m. Okay. Now what is n squared? n squared can be written as 4n squared. It is my Yeah. So now you can write this as 2k where you are calling k as your 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 what if 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 did not get a promotion 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 and it is 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. That means your negation p is true. Yes sir. You are proving that if not q then not p is actually possible of this. Okay. You have seen from the proofs tables that 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 contra positive approach? Let me go back and give you the same question in a slightly different way. Let's say there is a statement r which says if n square is an even number even integer then n is also even. Okay. Now try to prove this by using direct method. If you use direct method you will see that there are some bottlenecks. 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 2f then your n is under root 2f. Now we are clueless whether 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 also even. Isn't it? Yes sir. So this has a shortcoming we cannot use direct method so let us use contra positive method. So in contra positive method we will say let q be false. Okay. That is n is not even number. But n square is even. Okay. Wait, wait, wait. So let n be odd by the law of x2 in middle if something is not even it has to be odd. So let n be 2m plus 1. So n square will be 2m plus 1 square. That is nothing but 4m square plus 4m plus 1. That is nothing twice of 2m square plus 2m plus 1. It is always 2k plus 1. That means it is odd. Correct? Yes sir. What will improve that n square is odd. Dash n. Which implies you have proved that it is false. Correct? That is the proof that negation is true. Correct? So by taking an assumption that negation q is true you have proved that negation is true. So we can say that this is true and hence this is true because both are equivalency. Are you getting my point how it works? Yes sir.