 In our previous video, we had talked about unions and intersections of sets and which how unions correspond with the word or and how intersections correspond to the word and when it comes to set theory. Now that we want to come over to logic, specifically Boolean logic, we want to see their counterparts. In logic, a conjunction is an expression that uses the idea of and and the symbol we use is actually an upward pointing arrow. If you want, you can compare this to the intersection symbol. It's just ours comes to a point as opposed intersection doesn't. I mean, to make a calculus pun here, a union has a horizontal tangent line while a conjunction is non-differential, all right? And so a conjunction is when you're using and so you put two ideas together by an and. And so consider that for a moment. Look at these three primitive statements. So P, Q and R, let P be the statement one is an even number. Yes, I know that statement's false. We'll play with that in a second, but just take it as a statement, right? False statements are still statements. One is an even number. Three is an odd number. Five is a prime number. So that's P, Q and R respectively. Respectively here meaning that the order I listed these is the same order I listed these. It's not scrambled up or anything like that. So this is P, this is Q, this is R. So if we were to consider the statement P and Q, and so that's how you read a conjunction, you just say the word and. Honestly, many of us will forget the word conjunction unless, of course, you saw that schoolhouse rock song, you know, conjunction, junction. Anyways, P and Q, it means you put the two statements together via the connection, the connective of and. So P and Q means one is an even number and three is a non-number. That's how you read it. Now, how do you interpret it? When it comes to Boolean logic, a conjunction is only true if both primitives are true. So if the first statement was true and the second statement is true, then the conjunction, because the conjunction is itself a statement. While P and Q were simple statements, P and Q is a compound statement. But as it is a statement, it is either true or false. And the truth value of the conjunction will be determined by the truth values of the primitives. And that's what this right here is supposed to indicate to us. If your two primitive statements are both true, then the and, the and statement, the conjunction will itself be true. So for example, if you promise your children that you're going to go to the store and buy them ice cream and cookies, if you don't buy them ice cream and cookies, then you lied to them that because your statement was false. If you go and buy cookies but not ice cream, then you still lie to them even though you give them delicious ice cream. Or conversely, if you buy them cookies but not ice cream, you've also lied to them that statement was false. The only way your children will be satisfied in this situation is if you get cookies and ice cream, both have to be true for your statement to be true. And that is what this is saying here. The conjunction is only true when both primitives are true. If one primitive is false, then the conjunction is likewise false. And of course, if both primitives are false, then the conjunction was likely false. If you don't keep both of the things you promised, then you're a liar. And that's how we should interpret these conjunctions. So then pivoting back down to these statements we were looking at a moment ago. P stands for one is an even number, three Q stands for three is an odd number, and R stands for five is a prime number. If I look at the statement P and Q, remember, this means that one is an even number and three is an odd number. This statement, one is an even number and three is an odd number is itself a false statement. Why? Because it's a conjunction. It's an and statement. Ands are only true if both statements, the first and second simple statements are true. But this one is a false one is not an even number. Well, you know, three is an odd number. This is true. And so we currently have a false and true scenario. So the first one was false, the second one was true. And so the and statement becomes false. This is false because the first thing is false. Likewise, if you look at the statement P and R, what would P and R be? P remember is one is an even number and five is a prime number. Well, the statement R is a true statement. The statement one is an even number is false. So when you conjoin a false and true statement via an and that makes it a false statement for the same reasoning. All right, both primitives have to be true for a conjunction to be true. On the other hand, if you take the conjunction R and Q, that would be the statement five is a prime number and three is an odd number. Well, five is a prime number. That's true. Three is an odd number. That is true. And so with the and statement, since they're both true, that makes the conjunction a true statement. The statement, which is compounded by the word and here's that's how you connect two statements together, its truth will be dependent upon the individual truths here. Since they're both true, the whole statement is true. An and statement is only true when both primitives are true. That's what we need to remember about and now let's look at the counterpart of a conjunction, which is actually a disjunction. And if you want to make a schoolhouse rock about that one too, you can do it, you know, disjunction, junction, what's my function? It rolls off. It's the same rhyme there. A disjunction conveys the idea and logic of an or much like we did with unions with set theory. Disjunctions are about or and the symbol is very simple, very similar as well to unions. The symbol for a disjunction, it kind of looks like a V, much like how for a union, you have a U and the same the same joke about calculus applies right here. The union symbol is going to be smooth. It has a horizontal tangent line as opposed to the disjunction. It's going to have a non differential point at that cusp right there. And so it looks like a V and it carries this idea of or where or means at least at least one is true. They can both be true. Both is okay. That's a possibility. No problem with both being true. And so a disjunction works in the following way. If one of the primitives is true, the or statement is likewise true. So if like the first primitive is true, then this is going to be true as well. So let me give you an example of this. Let's say you're trying to make a promise for your kids here and which case, oh, let's I'm going to go to the store and get ice cream or cookies because you're such great kids or something like that. When secretly it's just you want ice cream or cookies, but you know, you make the kids feel like they earned it. You're going to buy them anyways. Anyways, so let's say that this is the promise you made them. Well, if you come, you'll go to the store and you buy the ice cream, then that means you were telling the truth. The kids will be happy about it. Conversely, if you instead buy the cookies, but not the ice cream. So the ice cream is false, but the cookies is true. You still told the truth. Everyone will be happy about it. Now, if the unfortunate situation is you go to the store and you don't buy ice cream and you don't buy cookies, right? Both statements are false. Then that actually means that you lied to the children and they will be extremely upset about you. They might not even let you inside the house. But of course, there is a fourth possibility. What if you go to the store and you buy cookies and you buy ice cream, right? What if both statements are true? Well, you still kept that you still were telling the truth, right? You bought cookies or ice cream, you got both. That was a possibility and they'll be ecstatic because you got both cookies and ice cream, right? So in the situation of both, that'll be true as well. Or does not mean in logic what it often does in English words, an exclusive thing. Or by default is inclusive of war where both statements could happen at the same time. So a disjunction is true if at least one of the primitives is true. If they're both true, that's also true. The only way a disjunction can be false is if both of the primitives were false. Now, conjunctions work the exact opposite. For a conjunction, sure, if they're both true, you'll get true. But if one of them is false, so if you get like true and false are false and true, of course, if they're both false, then you're going to get false in that situation. So there's this duality between conjunctions and disjunctions. A conjunction is only true when they're both true and a disjunction is only false when they're both false. But just think of it as and or reasoning. So let's look at our primitives, P, Q and R again, where P is the statement, one is an even number. I know that's false. Let Q be the statement, three is an odd number. I know that's true. This time we're going to tweak R a little bit. This time R is going to be six is a prime number. The reason I changed is now this is a false statement, five is a prime number. That was a true statement. So I did want to change it a little bit because disjunctions and conjunctions work a little bit different. So look at the compound statement P or Q. So P was the statement, one is an even number. Q was the statement, three is an odd number. And our V symbol here means or. And so that's our compounded statement, the disjunction. So from what we've seen before, I guess I just hit it. There we go. From what we've seen before, P is a false statement. One is not an even number. Three is an odd number. And since this is an or statement here, we end up with false or true. This compounded statement is in fact true. So you might be like, okay, either one is an odd number, excuse me, one is an even number or three is an odd number. That's a true statement. Even though I know one of them is false because the other one's true, that makes the compound statement true. And similarly, if you took the statement R or Q, that's also a true statement. Or excuse me, R is six is a prime number. That's a false statement. Q, three is an odd number. That is a true statement. So false or true is true by the same reasoning as above. Now, consider the statement R or Q. So one is an even number, which is not true. Or six is a prime number. That's likewise not true. So you have a false or false happening right here. False or false is actually a false statement. Because both primitives are false, that means the compound statement is also false. And so from this video, I just want you to be able to learn how you can compute the truth value using these two connectives, the conjunction to form an AND statement and a disjunction to form an OR statement. And we'll do some more calculations with these new logic operators in the future.