 Compound statements. Did I unshare this thing? No sir, it's nice. What are 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. If you look at these statements, 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 function which we call here as the AND connector. Now this connector is basically represented by this symbol in logic and such a statement will be called as a conjunction. So when you say p and q mathematically or symbolically it is represented as p up arrow 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? Nothing. No sir. 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 because it's a grand trade and trade or trade or all those kind of things. Okay. The purpose of simplicity you can assume it to be the multiplication. Okay. Now how does a truth table of AND look like? So something called truth table, what is a truth table? Yeah, 1100. So basically it is a kind of a tabulation of what is the outcome of a compound statement for different, 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. Okay. 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, sorry, p 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 when your conjunction is going to be true, when your conjuncts both are true, when your both the conjuncts are true, then only 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. Here 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. This was done representing the symbol down arrow. Let me give you an example. Let's say I give you a statement Venkat is smart and lucky. Sorry. Okay. Now here, you may be both. Yes sir. Two types of OR which we normally come across in our English language. One is called Exclusive OR, another is called Inclusive OR. What is Inclusive OR? The statement I gave you, this is actually an Inclusive OR because you can be both. What did I say? Nagma is dead or alive. It is exclusive OR because she 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. Not counted. Yes. So there you cannot include both the situations in the statement, but here you can say yes. Venkat can be both smart. 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, Exclusive OR, I will not talk about Exclusive OR. Okay. Is that fine? Yes sir. And again, please look at it every OR as a connector. For example, I am going to watch the movie Seetha or Geeta. But here OR is not really connected. Okay. What about the truth table of this Let's talk about it. It will be true, true, true, false. False only when both are false. The truth will give you a truth. True false will give you again a truth. False true will give you a true and false false will give you a false. That means only when the component or the constituent statements are false, then only your disjuncting will be called as false. Yes sir. Something very important here What is the negation of a disjunction? Sorry, conjunction. I claim that the negation of a disjunction The negation of a 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 you 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, not p, not q. Yes. No sir, 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 output. How many inputs do you have for p and q? Remember it's always 2 to the power 2. Oh, that way. Yeah. So if there are p, q, then you'll have 8 inputs. Let's say true, true. So what will be this? False, false. This will be false. False. False or false which is false. Correct. Thank you. False. This will become false, true. False, true. False. It will become a true. This guy will be true. That will become a true. True or false, true. False, true. True, false. True, false. True and false, false. False, false. Then true. True. 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. This will be true. Oh, yeah. Correct. No. So as you can see, 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 do it by truth table. One second here. This is some one second copy and I'll take screen chart. Yeah, yeah, please. Done, sir. Okay. In a similar way, following the modern law only, I can say negation of a disjunction is logically equivalent to conjunction of their respective negations. Okay. I'll leave you as an option. Please prove it using TT. Is that fine? Yes, sir. Okay. Now comes the important part. So before we were doing any activities, but now comes the serious part of this chapter which is called implications. Implications are also you can say compound statements, but here the connectors are not the and or the or, but it is something like this. Let me give you an example of an implication. By the way, implication is also called conditional statement. Typical example of implication. 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 and then I will not go out. Okay. So you can say this is one statement and this is another statement Q. Okay. And if he then Q is symbolically represented as P single direction arrow Q. Okay. Where P is sometimes called the antecedency. Okay. Many book will call it as hypothesis. Many books will also call it as the premise. Okay. Pre-mise. Oh. Q is called the consequent. Precedent. Oh. Consequent. Okay. Yes. Some 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 antecedent to that. Okay. And remember, it is the direction of the arrow is only one side. It doesn't go the other way. What does it mean? It means that P is the sufficient condition for Q but Q is the necessary condition for Q. What sir? I'll give you an example. Yeah. Write it down. P is sufficient condition for Q but Q is a necessary condition for P. Sufficient means it is, it is sufficient that if P happens, then Q will happen. Right. But if Q happens, P may or not, are you getting my point? No sir. Let me give an example. If it is a Sunday, okay, then school will be closed. Okay. So, if somebody tells you that, hey, today is a Sunday, then what, what conclusion can you draw about the school being open or closed? Pakka closed. Definitely closed. Pakka closed. Does it mean it has to be a not necessarily? It may be closed because of some, right? It may be closed because of some, you know, a riot in the town. Coronavirus. It may be closed because of Coronavirus. It may be closed because there's 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. So, you can also read this as P implies Q. One second sir, one second. 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. And even a final exam is coming for like one mark, one to two marks, that's it. Please also one question will come, not more than that. But that's, there'll be a short, short question, right sir? Sure sir. We'll definitely get four marks here. Yes sir. Okay. Yes sir. I'll return. There's no need to write down and I'll be sharing this with you. Okay. Oh, seriously then chill, I'm 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. So, this is an implication. There are certain terms that we need to understand. If you write this rule down, we call it as convert from this implication. Okay. Now, remember if P implies Q, P may not be true. That means if 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. Like two minutes. Two minutes. We can write it down. Oh yeah. So, if X died because of coronavirus, Q implies P, X died doesn't mean he died because of coronavirus. Okay. I'll give you a very related example. If a number is divisible by, is divisible by, what has happened to my spelling? Yeah. 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 proves 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 at that time. Why are we proving the same thing, Ulta and Sita? You know, two times because of the fact that if an implication is true, its 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 this statement? Tell me, what is the converse also? Okay. Devota 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. Yeah. 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. Okay. So the answer to that is yes, contra positive, but we'll see that through 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 if true or not? True. True. If this is true and 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. Yeah. Yeah. False. True. True. Yeah. It's double negative becomes positive. Doesn't happen in the other thing, da. Yeah. I'll 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. Now, tell me when does the husband make a good husband? When he keeps up his promise. Of course, when he keeps up his promise. So let's say he makes a promise. That is true. True. It takes his wife to Singapore. That is also true. Is he a good husband? Yeah. Yes. 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 husband. Bad. Bad means false. Okay. Yes, sir. If he doesn't get a promotion, poor fellow, but still takes his wife to Singapore. Yeah. Nice guy. But God's husband, right? So again, write it true. 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'll get into IIT. Now, if you studied and you got into IIT. Peace. There's no problem, right? Peace. But if you studied and didn't get into IIT, then definitely you'll come and shout at it. Sir, you said you studied, but I still hadn't. So if the antecedent is true, but the consequent fails, then only you'll come and shout at it. Then only you'll raise a question. But if you study and let's say you cleared IIT, that means you're still happy, right? Because antecedent here is false. That doesn't mean your consequent cannot happen. Just like your school. If your school is false, it may not be a Sunday. Oh, 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'll not get into IIT. Yeah. Now, I would like you both, not both, all of you here. I actually thought two people are funny, so two people are sitting 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 verify whether the text is written as what we got over here. I want you to give me the truth table for this. When you're 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 three so that we can check which of the three that we had discussed, the inverse, the inverse and counter positive is logically equivalent to the implication. Okay. So, sir, will this table just be the bottom to top for the above table? We'll check. I don't remember the exact answer. Independent how you start and stuff, but okay. Can we discuss now? Wait, sir. Q implies P. Can we do it together? Okay, sir. Yeah, that's better. So, just the truth table off Q implies P. So, we did the reverse. Promotion takes his wife. Who has been? True. Yeah. Didn't get a promotion, takes his wife. Who has been? Yeah. Yes, promotion doesn't take his wife. Okay, promotion doesn't take wife. Not P implies not Q. So, we did it like this. False, false, right? Yeah. False, true. True. True. False, true. False. False. False to true. False to true. Oh, yeah. Correct. False. False. True, true, true. So, you took that antecedent thing as Q, right, sir? No, no, no. I'm talking about this guy. Oh, okay. Okay. 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. 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. 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, true. Yeah. False, true. True. False, true. Yeah. Good guy. Okay. Now, you can see here very, very clearly that 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 logically equivalent. So, many books write it like this. Many books write it like this. Both are fine. It's logically equivalent to saying if not Q, then not T. Okay. Yes, sir. Is that 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 once again. 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. This is your husband's promise and then this is your wife speaking to Singapore. For this guy. Okay, sir. One second. For my future reference, I hope no one will mind. Anyway, you are giving this to us. One second, sir. One second. Okay. Done, sir. That's an E by the way. Yes, sir. Thank you, sir. What about the negation of this? Or the negation. Negation of this. I claim that this is the same as saying P happening and two not happening. Okay. Let's verify this. How, sir? How, how? Let's verify. Okay. Okay. P, Q, P implies Q. Negation of P implies Q and 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 P and not Q? Not Q, false, right? Yeah. Correct. Yes, sir. Yes, sir. So far, false, false same. Let's see two false. That has been. Yes, sir. False. Negation of it is true. True. True. True and negation of false. True and true, that is, which is true. Okay. Now, false, true. Nice guy, damn nice guy. Nice guy, this one. False and false. False and false. This is false again. Yeah. Then finally. False and false. False and false will be true and then false. True. This will be true. True, true, false, sorry. True, false, false. Do you see that there, 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 IK. I studied, but I did not get into IK. 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.