 definitely one question comes in J.E.Main from this chapter methodically, in the era of the geometry was evolving, I mean in the era of those mathematicians coming up, Aristotle and all was working on geometry, there were a group of travelling teachers which they called sophists, Now these guys, they had a reputation of arguing for any point and 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 should 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 Reputation of the Sophists. That book was called Reputation 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 statements or what are propositions. Have you done this chapter in school? Yes sir. Okay, so you tell me what is the meaning of a statement, mathematical statement or a proposition? It's not vague. It's not vague. What about you? 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 expressions. 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. So let's say like 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. To give you some example, 2 plus 2 equals 4 is a statement because you can say true for it. Why is the rational number? That's also a statement because you can say false for it. Yes or no? Yes sir. 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. All rules are white. Is it a statement? Yes sir. Okay. So I will not waste time on this. These are something which we have already done in the school. So we represent a statement by a symbol P. P stands for proposition. Okay. And whenever we have to write a statement, we write P colon and then we write that statement. Right. So if I have to say a statement that Venkat is happy. Okay. I will 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. Hello. The negative of this statement. So if I say Venkat is happy is a statement P. What is the negation of 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 can write anything in the exam or it is all 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. One is called a universal quantifier and the other is called an existential quantifier. Okay. What are the universal quantifiers? Have you done this? No sir. 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 there exists or some. Okay. These are called existential quantifier. Right. So I'm sure you would have used this simple. For all. Or there exists. Okay. So 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 which 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. 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. You can say there exists a rose which is not white. Or you can say it in various ways. Some roses are not white. They are 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 a universal quantifier All roses are white. 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 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. Okay. So let's say I give you a statement. Everyone in France, let's say I give a statement to everyone in France. What is the negation of? 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 do not speak French? That will be wrong, right? Because 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, I 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. Oh, when language was being created, there was, there was actually a meaning found out in the utterance. And then it was made a sentence. So at that time, like for us now, gouligouligou didn't make any sense. But that time when it was being. Right. For you now, very German will not make sense. Right. But now there's a pattern in which they are speaking it. So for them it makes sense. But don't worry about it. Okay. Okay, sir. Yeah. 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, there exists a cat which there is glasses. Okay. Okay. What would be the negation of that? All cats bite. All cats bite. 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 whether we have a question based on this. In our previous J exams. 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 Venkat. They have given the speaking role to you. Other than you. One Dheeraj was there. That also has stopped coming. I don't know why. So Niyati says something else. Venkat you said C, right? Yes, sir. Okay. What about you, Amla? Dheeta? Dheeta? Say something. I talk a lot, sir. No, no, no. That's fine. You are at your normal. Screen frozen. But mine is working, right? Read the question. Question says there is a rational number which is greater than zero. It starts with an existential quantifier. Isn't it? Yes. I think C. That means you have an existential quantifier here. Okay. Then read carefully and tell me which of the options is correct. It says have a universal quantifier. Tautology. What's tautology? I'll come to that. Okay, sir. Tautology is a statement which is always true for whatever component values that whatever value the component statements are. 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. Okay. This is something else. This is actually complicated. 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. Now 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. Good, Niyati. Okay. I have to deal with this kind of no problem. And let me tell you one problem and definitely