 Welcome back and welcome to the first of several screencasts we're going to look at where we construct truth tables for logical statements. So let's start with something that's kind of simple and it's just an English sentence. Okay, it just so happens that Apple computer updated its operating system yesterday. So it kind of inspired this sentence here. Kind of a lengthy sentence so we're going to parse this out. It says that Apple will either update its operating system today or it will not announce a new line of computers. So in real life we might be interested in when whether this statement is true or not. So notice that the statement here, this giant sentence, first of all it is a statement. It's a declarative sentence that's written well in English that is definitely true or definitely false. But the truth value depends upon the statements that make it up. Let's first of all highlight the individual statements that make this up. You can see that there are two of them. One of them is Apple will update its operating system today. Let's just factor out the word either just for now and just kind of parenthesize that. Apple will update its operating system today. That's one sentence that makes up this larger sentence. This is what we call a compound statement. It's a statement that's made up of smaller statements that are joined together by things like AND and OR and so forth. And the second sentence you see in here is this one and we're going to think about that a little bit more carefully. It will not announce a new line of computers. So that's two statements that make up this larger statement. And of course they're joined by OR. So this is what in your technical terminology would be called a disjunction, an OR statement. Let's think about these statements here. Let's just make things shorter. This is kind of a lengthy statement. Let's call this first statement Apple will update its operating system today. Let's just symbolize that with the letter P. So the letter P will stand for the entire English text Apple will update its operating system today. Now the second statement, it would be tempting to just take this whole thing and call it say Q. But I think that we want to make this statement as simple as possible. So I'm going to take the word not and just kind of factor that out for a second. The word not there indicates a negation of something. So let's make the second sentence that's involved in this statement to be as simple as possible. And let's make that sentence Apple will announce a new line of computers. So let's call that statement Q. And so again that's the statement that Apple will announce a new line of computers. Okay. So what you're reading in the sentence here, the thing that I've highlighted, the entire thing would be not Q or the negation of Q. So we use that little symbol to indicate negation. So what we have here, if we're going to translate the entire English sentence to symbols would be P or not Q. And we would symbolize that as P with a little V looking symbol. P or not Q. Okay. Now that's a compound statement notice that involves two statements itself P and Q. And so whether the entire statement is true or not depends on whether P is true and whether Q is true. And that's where the truth table comes in. So a truth table, which we're going to do right here is a table that records all the possible truth values of P and Q and follows the consequences and see to see whether the large statement that we're interested in is true or false under each condition. So since there are two statements here, we want to look at all the possible combinations of those truth values. And there are four of those. Let's just list those out. I'm going to make a column for P and a column for Q. And in each one I want to list all the possible eventualities of whether P is true or Q is true. We could have P and Q both true. That could happen. We could have P true and Q false. We could have P false and Q true and we could have them both false. So there are the four possible combinations of truth values of P and Q. If I were tracking that original English sentence to see whether it was true or a lie, I would want to see like, okay, did Apple actually announce an operating system today? And did it actually announce a new model of computer today? Whether this statement is true or false depends on whether the little statements are true or false. And so we're just keeping track of those possibilities. Now, we're going to build this statement piece by piece and go in. It's almost an algebra problem where you do what's in the parentheses first. So we're going to start small by looking at the negation. And I want to see, first of all, make a column that tracks the truth value of the negation of Q. And then once I've tracked the parenthesized expression, I'm going to be able to determine whether the entire statement is true. So I'll make a column for that as well. So let's first of all fill in the column for not Q. Not Q, the negation of Q is just the logical opposite of Q. So whatever Q is true, not Q is false. Whether Q is false, not Q is true. Okay, so here is the column for the negation of Q. And now I need to put together P, not Q, and join them with an or. So I'm just going to kind of highlight the two columns that are of interest to me here. Now, you've learned in your reading that a disjunction here is true when either or both of the statements involved is true. But if both of those statements happen to be false, then the entire disjunction is false. It's an or statement. It's the same way you use or in regular English. So let's look at each line. In line one, P is true and Q is false. Since P is true, that makes the entire disjunction true. In the second line, both P and not Q are true. And so that makes the entire statement true. In the third line, P is false and not Q is false. And so that makes the disjunction false. And finally in the last line, P is false, but not Q is true. So we have a true down here. Okay, so there is the final result. I now know exactly when that statement about Apple updating its operating system and not announcing a new computer is true or false. So that's a truth table. And we're going to in future videos move on to more complicated truth tables to analyze more complex statements. Thanks for watching.