 Welcome back again to this series of screencasts on TruthTables. This is part five, and we're going to take a look at a sentence that is a lot simpler than the ones we have seen in the last couple of screencasts. We're focusing in here on a new connective device here for statements that you learned about in your reading. Apple will announce a new computer model if and only if, and I underline that, it announces a new line of music players. So the if and only if right here is what you learned in your reading as a bi-conditional, or more correctly the statement you are reading here is a bi-conditional statement, not a conditional statement, which is like an if then, but it's really a pair of if thens. A bi-conditional statement is true when both of these statements that adjoins are true or when both the statements that adjoins are false. In other words, it wants the two statements that adjoins to have the same true value, whether it's true or false. They can't differ from each other if the bi-conditional is going to be true. A bi-conditional statement, in other words, is just another way of saying that two statements mean the same thing exactly, like today is July 4th and today is Independence Day. Those two sentences are the same, really they look different, but one is true only if the other is true. So let's parse this out. It's pretty simple to do here. The first statement that is in this bi-conditional is Apple will announce a new computer model. And the second one is Apple announces a new line of music players. And so to put this into symbols, let's call the first sentence P. Again, there's nothing to break down in this first sentence here. There are no ands, ors, or nots, or if thens involved. And the same thing is true for this one, so we'll just call it Q. So there's only two basic statements in here. We'll call them P and Q. So this is saying P, if, and only if, Q. Or in symbols, we will write P, and then kind of a double-headed arrow that points both left and right. P, if, and only if, Q. Sometimes you see if and only if abbreviated I, f, f in the literature. So let's set up the truth table for this. Again, the thing to remember is that P, if, and only if, Q is true in two conditions. That's when the two statements that are being connected are both true or when they're both false. If either of these two statements differ in truth value, then the entire bi-conditional statement is false. This is a two-statement statement, and so we're going to set up four rows. Let's list out the truth values for P and Q. True, true, true false, false true, and then false false. We're only going to have one line for this, and this is just going to copy what you learned about bi-conditionals. In the first row, the two statements are both true. They have the same truth value. And so the bi-conditional is true. In the second statement, one is true and the other isn't. That makes the truth, the bi-conditional statement false. In the third line, again, one is true and the other isn't. And so that makes the bi-conditional statement false. And the last one, both statements are false. That makes the bi-conditional true. They share the same truth value. In other words, P and Q really are the same statement. When one's true, the other's true, and when one's false, the other is false. So thanks for watching.