 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. Do you remember that real number chapter you had in series 2 class 10? Prove that route 2 is irrational, amazing. Prove that route 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 and Q is false. Why have I chosen this actually? You contradict yourself. See, this was the only condition when your written statement was false, right? Remember the truth? Yeah. Husband gets a promotion and doesn't take his life for the foreign trip, for the Singapore trip. This was the only situation when your written statement 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 involved. First you assume that P is true and Q is false. By assuming that you are indirectly assuming that your 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? Okay. Yes sir. Might be. A, don't give such a cold response. What is, what is not clear in this aspect? It's it. Bandwidth is not matching. One second. Bandwidth is not matching. If P is true and Q is false, yeah. Yeah. Okay. Yeah. Fine. Got it sir. Can you do the same question which I had given? Let's say I 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 zero, where X is your real number, then X is zero. Validate this by all the three methods. Direct method. Contra positive 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 X square plus four, which is that complex thing, but then you're given the condition. Then we'll discuss it. Yes sir. I've done it. Not sure if it's right or not. Yes sir. Can we discuss? No sir. Yes sir. I did something. I'm not sure. We'll start with contradiction method because we haven't done this yet. So in contradiction method, what are the steps? So first we'll assume P. So this is your P by the way. And this is your Q. False. It's better to write them very, very clearly. P and this is your Q. So in contradiction method, let P be true. But Q be false. And yeah, correct. You should use but not. That means you're saying this guy is zero and this guy is not zero. Right? Yes sir. Yeah. That means you're saying X into X. X into X square plus four equal to zero. Since this guy is not equal to zero, this guy should be equal to zero. Now we know that. Now we know that. If X not equal to zero. Real number X square plus four will always be a positive quantity. Isn't it? Yes sir. That means if the product of X with a positive quantity is zero. That means only X could be zero. And this is not zero. It's just a contradiction. At one place you're saying it's zero. And another place you're saying it's not zero. Yes sir. That means my assumption is that the assumption itself is false. Okay. Correct. That is, that is, P implies Q is true. Isn't it? Yeah. Because the assumption is that P implies Q is false. And you have proved that false itself is false. It means it is true. False itself is false. Yeah. Basically what you are saying. Yeah. You are saying negation of a false, right? 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. Okay. Do direct method now. This is done. This is what I meant by saying solving method. Basic solving you have to do. Yes sir. Sir, counter positive done. I thought you were doing it right. No sir, no sir. Do it anyway. Okay. Let's try it right 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 scroll up to the direct method page one second? Okay, direct method is assume p is true and then prove q is also true. Done sir. Okay, we will discuss direct method now. In direct method what we say let p be true. That means let x cube 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 and 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. Okay. 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. No. Okay. Yes. No. This can be written as this. Okay. So if x is positive. Then x into x square plus four will also be positive because this guy is always positive. Okay. And if x is negative, then x into x square plus four will always be negative. Either of them implies that this term will not be zero. Yes. That means you have proved that p is false. That is your prove 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. 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 a city which is calling is not a city of D. D, D, D. I think D for Delhi. All of them are negation of p. So this is also one of these. Because they are asking not a negation of p. Yes, sir. Is that fine? Yes, sir. If p is a statement in which of the following is a tautology. Yes, sir. Is a tautology. A is a policy. One second sir, one second. My statement bro. F means a statement which is false. Okay. This is. Okay, one second. C. None of these. Is this a tautology? No, right? No, sir. Is this a tautology? No. Is this a tautology? If F is false then false or false which is bad. It's not a tautology. If p is false, sorry. Sir. It depends upon what is p, right? If p is false it will be false, the entire thing. Correct. So it depends upon p here. So this also often d will be correct, correct? Yes, sir. Okay. Let's skip the third one and then 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. Some printing mistakes seems to be right. Wait, sir. I don't know. P means not q. Okay. Like I didn't even get that. So. I just need these two. I think some printing mistake. Let's go to the sixth one. Equal means it is a tautology. P and true and false. No. Is this a tautology? Sir. No, sir. What about this? P. No, sir. Tautology. So it's correct. C. Is it the following statement? Q. That's q. No, no, no. Not p. D, sorry. D. P is correct. Yes, sir. Okay. This is the. Contra positive. Contra positive. Correct. Correct. Correct. Correct. Correct. Correct. Correct. Correct. Contra positive. Contra positive. Yeah. Next. Kevin's one. P and q are false. P. Sir, B. If both P and q are false. Intersection cannot be true. So this is wrong. Union can be false. False, right? Yes, sir. Is this correct? Okay. Say multi-choice or single-choice. Let's see. It may be multi also. Never know. J advance as this chapter. But this may actually include false. Okay, okay. False implies false. This is true. This is also true. Yes, sir. So B and q are false. Damn, bro. Both P. No, no, no. Gone. Okay. Same thing. We can do nine. P is two quills. Which of the following is not true? Yeah. P or A and B. This is false. False. That one, no, that one, no. This is basically the answer. This guy. So this cannot be answered. This is a. This is a. Yeah. So A and B are correct. Yes, sir. Complete this exercise. Okay, sir. Which one? What's that? 10 to 10. 10 to 10. 10 to 10. This is false q is true. This is true is wrong. No. This is. False. No. Yeah. This. No, no, no. This. 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. Oh, okay. He took his wife for the foreign trip. Right. Yes, sir. Yeah. Next. Converse of. Converse of the statement P. Oh, this one. Okay. What is that converse? Sir, finding contra positive is similar to reflection about origin. As is realized. Widyotha. Two steps in one. What is the answer for the 11th one? Widyotha and Advaita. Converse of this. While I was explaining it to you know what the converse is. Converse is Ulta. Yes, sir. This is called contra positive. Contra positive. It's called inverse. Yes, sir. Next. Contra positive. Okay. 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 influenced. Okay, sir. 12th one, right? 12th one, yeah. By the way, when you're writing, write as the question number and your answer. So, when I know it from your answer. Okay, sir. Next, which of the following is logical? Negation or negation? One second, sir. Okay, Widyotha. Sit down. Let's wait for others. One second. Advaita, what about you? One second, please. Let's see. Oh, sorry, sorry, sorry. I shouldn't have said that. Advaita, discuss correct? Let's discuss. So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, So, That's a P and not P, so 2 false, false words, 2 true, false, this is not the answer. Yes sir, so if A is not the answer, why will D be the answer? What do you mean by that? Contra positive, right? Contra positive, Contra positive is not Q, implies not P. Yes sir, yes sir. This is a AND symbol in between. 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 is again matching, this is matching, so D is the option actually. Clear, do you want me to do it separately in some different page or is it understood from here? Understood sir.