 Welcome back to another example of where we're taking a statement and deciding whether it is a tautology, that's a statement that's always true, a contradiction, a statement that's always false or something that's neither, which means that sometimes true, sometimes false depending on P and Q. So this one is saying that P or Q, I'm sorry, P and Q, if and only if, remember our biconditional statement here, not Q or not P. Now this seems false, it's on the surface of it. You're saying that P and Q means the same thing logically, which is another way to think about biconditionals as either not Q or not P. That seems wrong because, or seems completely false because P and Q is true only when P and Q are both true. And it's just saying that one or the other could possibly be false. So we're kind of suspicious about how the statement's going to go on the very outset. Let's take a moment to set up the truth table. Let's first kind of pick apart the individual pieces here. When I start making the truth table up, and look at the left-hand side of this biconditional, I think that should be maybe one column we have to make. The right-hand side's a little more complicated, so this is probably going to be a second column. This one over here should be third. Then I have to join them with an or, so that would be a fourth column. Then I got to connect the whole thing together, like a giant brace here, and call that column five. So through the magic of quick video editing, I've immediately set up this entire table here, including P and Q for starters, and here are our usual four combinations. Again, the point here is to break this complicated statement down into little atomic statements that are easy to evaluate here. Things that we just know, like P and Q. We know that P and Q is going to be true whenever both P and Q literally and Q are true. And that only happens here. All the other places, one of those two statements, possibly both, is false. Not Q is just the opposite of Q, so this is false true, false true, and not P is the opposite of P, so this is false, false true true. Next, I'm now working on the right-hand side of this bi-conditional statement. And here's the entire right-hand side, not Q, and that's an or, not P. So I need to look at not Q and not P, and basically ignore everything else, and join them with or, which means that if one of these two statements is true, then the entire or statement is true. And the first line, both are false, so that makes the or statement false. And in each of the remaining three lines, at least one of those statements is true, so there's my result. Now, let's recall how bi-conditional statements work. Remember, bi-conditional, a bi-conditional statement is basically an assertion that the statement that's on the left of the double arrow means the same thing logically and in formal terms as the statement on the right. And what we're gonna see if that is the case is that in each row where P and Q have a set truth value, the statement on the left and the statement on the right share the same truth value. They're sort of joined at the hip with regards to truth values. So let's ignore all else except the left-hand side right here. I'll just put a little l for left-hand side and the right-hand side, which is right here. Now, let's see what happens, okay? So in this case, the left-hand side and the right-hand side are different, so the bi-conditional is false. And the same thing happens in the second row too, the left-hand side is false, but the right-hand side is true, so that's also false. And actually, that happens in the third row and in the fourth row. So what we have here kind of confirms our intuition. We said that, look, I mean, P and Q being true means that both of these guys are true, both P and Q are true. There's no way that either one of these could be false, and that's indeed what we're seeing here. In no sense, and in no situation is the statement P and Q equivalent in any way to not Q or not P. So what we have here is a statement that is never true. It's the complete opposite of a, in some ways, of a tautology. So we call that a contradiction. Contradictions are very important for us in communicating in mathematics. They're very useful tools for us as we will see later.