 Most proofs in mathematics can be built around the conditional if A, then B. We say that A is the antecedent, and B is the consequent. So, for example, if this is Tuesday, then this must be Belgium. In the statement, this is Tuesday is the antecedent, this must be Belgium is the consequent. In a perfect world we'd always use the if, then construction for conditionals. We don't live in that world. And we often need to rewrite statements as conditionals. It helps to view the conditional if A, then B as follows. If you have A, then you have B. So, for example, let's rewrite as a conditional the square of an odd number is odd. Now our statement is claiming something about the square of an odd number. And it appears that we have the odd number. And so consequently, well actually antecedently, we have the square of an odd number. And the claim is that we also have an odd number. And this suggests that our simple statements are a number is the square of an odd number, and a number is odd. So rewriting this as a conditional gives us if you have the square of an odd number, then you have an odd number. Now while this is a perfectly good way to write the conditional, we might try to clean up the grammar a little bit. Well, we could start with the square of an odd number, for example 49, in practice we're more likely to start with an odd number and square it. And so our antecedent can be that we have an odd number and that we're squaring it. So rewriting this a little bit more simply, if a number is odd, then its square is odd. Another place where we might have a hidden conditional is in statements like all and every. Consider a statement like all x are y. This says that if you have x, then you have y. So this corresponds to the conditional if x, then y, or cleaning up the grammar a little bit, if something is x, then that thing is y. So let's try to rewrite as a conditional all prime numbers are odd. So again let's identify our simple statements. We have a prime number, a number is prime, and then we have that the number is odd. So remember all x are y is the conditional if x, then y, and so this gets rewritten as a conditional if the number is prime, then the number is odd. Another common phrasing is no x is y. It seems that we have no x, but it's useful to remember we want to make positive statements, and so it's easier to suppose that we have x. If we do, then since no x is y, then we know we don't have y. And so this gives us the conditional if x, then not y. So we could try to rewrite as a conditional no multiple of 100 is prime. And so our statements, a number is a multiple of 100, and the number is prime. So again remember the statement no x is y corresponds to the conditional if x, then not y, and so the corresponding conditional if a number is a multiple of 100, then it is not prime. Sometimes we have paired conditionals if a, then b, and also if b, then a. This pair is called a biconditional, and we could write it this way, but we won't. We could read this as a if and only if b, but we'd rather not. While we commonly do read it as a if and only if b, here's a useful idea to keep in mind. How you speak influences how you think. You should always speak of biconditionals as two conditionals if a, then b, and also if b, then a. So we might rewrite the statement a matrix has eigenvalues if and only if it is square. So let's identify the simple statements. A matrix has eigenvalues, a matrix is square. Since this is an if and only if construction, then as two conditionals, we have if a matrix has eigenvalues, then it is square. And then switching the antecedent and consequent, if a matrix is square, then it has eigenvalues. Given any conditional if a, then b, there are three related conditionals. The converse, where we switch the antecedent and consequent, if b, then a. There's also the inverse, if not a, then not b. And finally, the contrapositive, if not b, then not a. So for example, write the converse, inverse, and contrapositive, all prime numbers are even. So a good strategy, if possible, rewrite statements as conditionals. So let's take a look at our statements, and we can break it down into these simple statements. A number is prime, a number is even. And so our statement corresponds to the conditional, if a number is prime, then it is even. So this gives us the following. The converse is where we switch the antecedent and the consequent. So if a number is even, then it is prime. The inverse is what happens when we not both of them. So if a number is not prime, then it is not even. And the contrapositive is sort of a combination of the converse and the inverse. We swap antecedent and consequent, and we not both of them. If a number is not even, then it is not prime. And finally, one of the most important occurrences of conditionals. Every definition is a biconditional. Wait, that's an every statement, so we should be able to rewrite it as a conditional. We'll leave that as an exercise. And this does lead to a very good strategy for mathematical proof. Always rewrite definitions as two conditionals. For example, we might define an even number as follows. An even number is a number that is a product of two and an integer. So first, let's identify these simple statements. A number is even. A number is the product of two and an integer. Since it's a definition, it's also a biconditional. And so we have two conditional statements where we have our antecedent and consequent from these two. And so the two conditionals are, if a number is even, then it is the product of two and an integer. And also, if a number is the product of two and an integer, then it is even.