 Welcome back again to another screencast part two in a series of screencasts on making truth tables. In this truth table screencast, we're going to look at another similar statement that's slightly different than the one we saw before. Again, it has to do with Apple Computer updating its operating system. And it just says that if Apple does not update its operating system today, it will announce a date when the operating system will be updated. So we can sense that there are really two sentences, two statements going in to make up this large statement here. Let's highlight those first. The first one you can see here and note the word if, but the sentence that follows it is important. Apple does not operate update its operating system today. And the second sentence that seems to matter is it will announce a date when it will be updated. So let's annotate the sentence here and just pick it apart. First of all, the first word in the sentence really tips you off. This is going to be an implication or a conditional statement if, then. The word then does not appear, but you can kind of feel like it's implied right in there. And you can might as well just write the word into the sentence right where the arrow is pointing. And it will be fine. So we see the hypothesis of this conditional statement is right here. Apple does not update its operating system today. And the conclusion is it will announce a date when it will be updated. So let's pick this apart and symbolize the individual statements that make this up. First of all, the hypothesis. Notice there's a not in here. And just like in the previous screencast, let's just pull that not out for a second and go with the simplest possible statement. And so let's let P, the hypothesis here, be Apple does update its operating system today. And so the hypothesis, the entire hypothesis of this conditional statement is going to be not P. That's really the first highlighted sentence is not P. The second sentence you see here is it will announce a date when it, the operating system will be updated. There's no negations in here and no other ands or or statements. So this sentence by itself is as simple as possible. Let's call that Q. Okay, so what we have here is the statement if not P, then Q. And we would symbolize that by if not P, might as well parenthesize that to be safe, then Q or not P implies Q. And that's the statement we're going to work with in a truth table. So let's make this truth table here. And again, we have two little statements that make up this entire statement. So just to quiz your knowledge of how to make truth tables, how many rows is the truth table for this statement going to have? Is it going to be two rows? Will it be four rows? Will there be six rows? Or eight rows or something else? None of the above, none of the above. So take a moment to pause the video and select the answer that you think is most correct. Okay, so the correct answer here is we're going to have four rows in our truth table here because there are two truth values for P and two truth values for Q. And let's list those out. Again, we're going with the simplest possible things first. So just the basic statements. The negation and the implication will come later. Those four truth values are true and true, true then false, false then true, and then false then false. Those are the four possible combinations here. So let's set up the truth table here. As we saw in the last video, we want to go as simple as possible. Let's do this in the parentheses first and make a column for not P and just to keep a track of its truth values. And then we'll have the next level up will be actually the final statement we're interested in. So not P implies Q. Now let's fill in the column for not P. That is just the logical opposite of the values of P. So where P is true, not P is false. And where P is false, not P is going to be true. And now I want to fit this implication together. Now go back to what you know about implications. And we know that an implication is true in all situations except one. And that's where the hypothesis is true, but the conclusion is false. So it's pretty important right now to understand what the hypothesis of the statement is and what the conclusion is. I'm just going to go over and label those. The hypothesis of the big statement is not P. So I'm going to label that with an H and just kind of highlight it. The conclusion of the statement is Q. So that's over here. So although we read these columns left to right, really what matters for us in the truth table is going right to left, really. The hypothesis is here and the conclusion is here. So let's go through and write down the truth values for this conditional statement. When the hypothesis is false, remember, it's a little counterintuitive. But when the hypothesis is false, the entire conditional statement is true. That's the case in the first two lines. The hypothesis is false both times. So I'm going to have true, then true. In the third line, I have both statements being true. And so we've learned before that the conditional statement is true under those conditions. Look at the fourth row. The hypothesis is true, but the conclusion is false. And that's the one condition where I get a false implication. It was a lie. Why is that a lie? Well, let's think about what that says. It would say that not p is true. That would mean that Apple did not update its operating system today. But it also did not announce a date when it would happen. So that would mean that the original statement wasn't true. They were false. So there's another truth table with a little bit more complexity to it with an implication that has a negation on it. Still the same process though. So thanks for watching and wait for the next video. We'll reduce some more of this.