 Welcome back once again and now we're on part four of this series of screencasts on making truth tables. And this one we're going to see a new extra wrinkle to our statement. Let's look at the statement. Again, we're looking at an Apple computer. If Apple updates its operating system today, then it will announce a new computer model and a new tablet model. And so let's, again, starting with English, let's parse this and write it in symbols. So we see an implication about to happen here. Let's just kind of note this. If, and actually have the word then here, so that kind of sets off the implication nicely. And another word you see here that's kind of connective is the word and. Alright, so having highlighted those, let's go back and physically with yellow highlight the smaller statements that make this up. First of all, we're going to look at Apple will update its operating system today. There it is. If Apple updates its operating system today. Now notice in the conclusion part of this conditional statement, there's really two things. There's Apple will announce a new computer model and Apple will announce a new tablet model. Now the thing to notice and what's different here than the previous statements that we worked with is that now there are really three independently functioning sentences here. The first one here, which you might as well just go ahead and call P. This is as simple as possible. There's no negations or conjunctions or disjunctions or anything like that happening. P is the sentence Apple updates its operating system today. But in the conclusion side of this, there are two Apple will announce a new computer model. Let's call that Q and then Apple will announce a new tablet model and let's call that R. So R is not related to, it's connected to Q by the and but it's not related to Q in any sort of logical or English sense. Q and R different statements. Okay, so this is now an implication that has not two but three statements involved inside it. Let's write it down as symbols and then we'll think about how a truth table might go. So it's an if then statement and the hypothesis is P. So if P then and now I have Q and R and they're really joined by and so this is what we called a conjunction and we write that as Q and R. So here is our statement we need to write a truth table for and again the twist here is that now there are three independently functioning statements inside it. So let's see how that will work. Now first little quiz here. We're going to make a truth table for the statement. How many rows is it going to have? Is it going to have three rows, four rows, six rows, eight rows, or nine rows? So pause the video and and think about which answer is most correct. So the answer here is that they're going to be eight rows. Now why is that the case? We can write these out and we will but just think through it for a second. Each of the three sentences involved here has two states either true or false. One, two, three statements and there are two states each. So I could have one of two possibilities for P true or false. One of two possibilities for Q true or false and one of two possibilities for R true or false. So the number of possible combinations is the product of those two times two times two that's eight. To list these out I'm going to need considerably more vertical space here. So I'm going to make three columns for P, Q, and R. And let's first of all list out all eight of those possibilities. This is not so hard to think about if you ignore R for a second. For P and Q we've already seen the four possible combinations of truth values and here they are. Now I'm just going to make a direct copy of these four rows down below. True, true, true, false, false, true, and then false, false. And I'm going to put a little dividing line for right now between those first four and second four. Now for the first copy I would like R to be true. And for the second copy I'm going to set R to be false. So since the third entry here is different from the first four and the second four these are eight truly different truth values. And here they all are. So with that let's go ahead and start building the truth table. Again just like an algebra we want to do what's in parentheses first. We want to parse the sentence down to its finest possible terms first. And that's to look at this conjunction here Q and R. So let's make a column for Q and R. Now we haven't seen a conjunction lately. A conjunction and an statement is true only when both statements involve are true. So I'm going to look at Q and look at R. And when both Q and R are true the conjunction, the and statement will be true. That happens in the first row. But it doesn't happen in the next row because Q is false. Does happen in the third row. Does not happen in the fourth row. And now let's continue. In the fifth row, now we have a fifth row, true, false. That makes the conjunction false. Both statements need to be true. It's just the same way you use the word and in regular English. In fact, I might as well go ahead and put false for the remaining three rows because notice that R is false. If R is false then Q and R can't possibly be true. And now we're up to the top most level of the statement. And that is to think about the implication if P then Q or R and R. Now to build that column I need to remember once more how a conditional statement works. A conditional statement like this is true in all cases except one. That's when the hypothesis is true and the conclusion is false. Let's label the hypothesis. Here's the hypothesis right there. And then here's the conclusion right there. So I now just kind of ignore the Q and the R for a moment and focus only on the hypothesis and conclusion. In the first line the hypothesis is true and the conclusion is true. So the entire conditional statement is true. In the second row the hypothesis is true but the conclusion is false. So that makes this false. In the third and fourth rows notice the hypothesis is false and that makes the entire conditional statement true. In the remaining rows I have true, false and that gives me false here. True, false that gives me another false. And in the last two rows the hypothesis is false so that automatically makes the conditional statement true. So there is, and we might as well go ahead and dash this out just to keep track. There is the final result of the truth table for this statement. If P then Q and R. So again the process of setting up this truth table is the same except for one twist when I have an additional statement thrown in I'm doubling the number of rows in the truth table because it throws another true false value into it. But we still track the consequences once we look at each possible combination of truth values. I've got one more screencast in this series on truth table so stay tuned.