 Welcome back, and this is the third part of a series of screencasts on how to make truth tables. So in this video, we're going to ratchet up the complexity of our sentence we're working with a couple of notches here. Whenever Apple updates its line of music players, it either announces a new computer model or does not announce a computer model. Let's go in here first and highlight the connectors in here. You can sense that there are English sentences here, smaller statements that make up this large statement. But let's highlight the ways that these are connected together. First of all, notice that either or here, we're just going to join those together. This is, there's going to be a disjunction in the statement somewhere. We see a not here, which is going to be a negation. And then the word whenever, you know, let's kind of highlight that up here. One of the things you learned in your reading is that among the several ways to phrase a conditional statement is to use the word whenever. So this is actually like an if, we can replace that word whenever with the word if. And those are synonymous in the logic language. So now that we've highlighted the connectors here, let's highlight the sentences that are being connected. First of all, Apple updates its line of music players as one. Another was, Apple announces a computer model. And another one is, does not announce a computer model. So I'm going to leave the not out of this and just highlight this sentence here. Now what you notice is that this sentence is the same as the previous one. So although there are three pieces of text that I just highlighted, there are only two statements here. One that says Apple updates its line of music players. I'll just call that P. And the second one that says Apple announces a new computer model. So that's Q. Now let's put all those together. And what this is saying is if P, let's start here if P, I'll write it in English, if P then, the word then doesn't appear. But again, you can kind of put it right in there, the word then. Then either, and I'm going to write a parenthesis here. We don't write parenthesis in this English sentence, but just to help us out. Either Q or not Q. So that would be sort of a halfway point between the English statement and logic. And now let's translate this fully into logic symbols. It would say P implies, or if P then, Q or not Q. And that's the statement we are looking at up above, short handed out in logical symbols. Before we move on to the truth table, this statement here is kind of weird, isn't it? Q or not Q. If Apple updates its line of music players, it'll either announce a computer model or it won't. It seems like this ought to be always true. It somehow will keep track of that feeling and we'll see how it plays out in the truth table. So let's set them up here. There are really, although there's a lot of action happening in the sentence, there's only two basic statements P and Q. And so just as before, we're going to have a truth table with four rows. Let's say P and then there's Q. I'm going to leave a lot of space here to the right though because we have a lot to build. So the four rows we could see here are the four possible combinations of truth values are coming up as you see. True, true, true false, false then true and then false then false. So the trick here is we have to really build the statement up piece by piece and there are a lot more pieces here than before. Here is perhaps the first piece the deepest inside the parentheses that we see and not Q. The next level out from that would be this part Q or not Q. So I'm going to make another column for it. Every time I reach another level of structure in the statement basically corresponds to a new column. And then finally I'm going to have the entire implication. So P implies Q or not Q. I'm going to keep my down that last column because that's what I'm really interested in. Okay so let's build this statement up piece by piece and we know that not Q, the negation of Q is simply the logical opposite of Q. So that's going to be false, true, false, true. And now I'm going to look at Q or not Q. The two statements involved in that disjunction are here and here. Now with disjunctions recall that we know a disjunction is true if one or the other or both of the statements involved is true. And let's just run down the four rows and see what happens. Okay and the first row Q is true so that makes the whole disjunction true. And the second one not Q is true so that makes this true. In the third row this one's true so the whole thing, whole disjunction is true and the last one this is true. So our feeling from before that Q or not Q ought to always be true does actually play out. This is always a true statement right this is always a true statement. How's that going to affect the entire implication? Well let's see because now I need to put the implication together. Now recall that an implication, a conditional statement has a hypothesis and a conclusion. The hypothesis of this conditional statement is P. I'm going to label that with an H just to keep track. And the conclusion is this whole mess right here. I'm just going to label that with a C. Okay so what matters here is whether the hypothesis is true and whether the conclusion is true. The conditional statement is always true. This thing is going to have all true except in one situation and that would be where the hypothesis is true but the conclusion is false. Now the thing to notice here is that that never happens. The conclusion is never false. And since the conclusion is never false we don't really have to do any work on this. The conditional statement will always be true. Sometimes the hypothesis is true and sometimes the hypothesis is false. But what really matters is whether the conclusion is false and that never happens. So this statement is always true. So this conditional statement, the large statement we had, as complex as it was, is a statement that can never be false. We're going to have a special word for that kind of statement a little bit later. But that ends the truth table and we see something we didn't already see about the structure of that sentence namely that it's always true. Thanks for watching and in the next couple of videos we're going to get even more complex but still follow the same process.