 Given the truth values of simple statements, we can compute the truth value of compound statements. So remember, definitions are the whole of mathematics. All else is commentary. And so here we're going to define when compound statements are true. So let x and y be statements. The negation of x, written this way, has the opposite truth value of x. The conjunction of x and y, written this way, is true when both x and y are true and false otherwise. The disjunction of x and y, written this way, is true as long as one of x or y is true. And the conditional if x then y, written one of two ways, is false when x is true and y is false and true otherwise. So for example, let x be the statement 5 is equal to 2 plus 3, and y be the statement 2 plus 2 equals 5. Let's find the truth value of x and y, the conjunction of x and y, the disjunction of x and y, and the conditional if x then y. So first, let's find the truth value of x, 5 is equal to 2 plus 3, and y, 2 plus 2 equals 5. And we see that x is true and y is false. Now the conjunction, written this way, so remember that's going to be true when both of them are true otherwise it's false. And since only one of them is true, then the conjunction is false. The disjunction, x or y, well that's going to be true as long as at least one of the statements is true. And so in this case, x is true and that makes the disjunction true. And for the conditional, the only time a conditional is false is when our antecedent is true but our consequent is false. And so here we see that our antecedent is true but our consequent is false and so the conditional if x then y is false. It's important to keep in mind this way of evaluating the truth or falsity of a conditional. So for example, let's evaluate the truth of the statement, if 2 plus 3 equals 6, then 4 plus 6 equals 0. So this is a conditional and our components, our simple statements, 2 plus 3 equals 6, 4 plus 6 equals 0. And let's evaluate those simple statements. 2 plus 3 equals 6, well that's false. And 4 plus 6 equals 0, that's also false. Since the only time the conditional if x then y is false is when x is true and y is false, then this is a true statement. And it's important to understand it's not that 4 plus 6 equals 0 is true, that's false. It's not that 2 plus 3 equals 6, that's also false. It is the case that the conditional if 2 plus 3 equals 6, then 4 plus 6 equals 0. This conditional is true. And this is worth talking about a little bit more. There's a few reasons for why we define the truth of a conditional the way we do, but one of the reasons is this. Remember that a statement could be true or false, but we don't know which it is necessarily. In a conditional, if the antecedent is false, we don't care. And what this means is that we only have to worry about the case where the antecedent is true. Expanding on that, suppose the antecedent of a conditional is true, then if the conditional is to be true, the consequent must also be true and this must hold every time the antecedent is true. And this leads to an important feature of mathematical proof. An example is not a proof. And that's because our statement has to be true every time the antecedent is true. So for example, let's try to evaluate the statement all prime numbers are odd. So again, it's helpful to rewrite this as a conditional. So let's break this down. Our simple statements are a number is prime, the number is odd. The statement all x are y is the conditional if x, then y. And so we can rewrite our statement as the conditional. If a number is prime, then it is odd. So let's try to evaluate whether the statement is true or false. So remember, it's a conditional. So the only thing that matters is what happens when the antecedent is true. And while we can't prove it this way, we might familiarize ourselves with the statements by considering an example of where the antecedent is true. In other words, where the number is prime. So if five is prime, then it is odd. That's our statement. The antecedent is true. The consequent is also true. And so this conditional statement is true. If seven is prime, then it is odd. And again, our antecedent is true. Our consequent is true. And so the conditional is true. How about the statement, if nine is prime, then it is odd. Well, here our antecedent is actually false. So it doesn't really matter what our consequent is. In a conditional, if the antecedent is false, then the conditional itself is guaranteed true. Again, an example is not approved, so we can't yet conclude that the statement is true. Since the conditional is true only if it's true every time the antecedent is true, we see if we can find a case where a number is prime, but the number isn't odd. And this may be one of the most difficult things about mathematical proof. A lot of the proof process involves trying to tear down your own argument to find every possible flaw in everything that you state. And this requires a degree of self-criticism that a lot of us aren't comfortable with. So here's an idea to keep in mind. If you don't find the flaw in your argument, someone else will. And in fact, you should treat it as if your worst enemy is going to be the one who finds the flaw. So you should make every effort to find any flaws in your own arguments before showing them to the world. So let's think about this. We want a number that's prime, but not odd. And it's easy enough to see that there is such a number. We might consider the statement, if two is prime, then it is odd. And here our antecedent is true, but our consequent is false. And so the conditional is false if the number is two, which means that our statement is false.