 Remember that any conditional, if a, then b, has several related conditionals. Could the converse or inverse be useful? In general, no. There is no relation between the truth of the conditional, its converse, and its inverse. However, the contrapositive has the same truth value as the conditional, so if we can prove the contrapositive, we simultaneously prove the conditional itself. The contrapositive is useful since it gives us a different thing that we can prove, and one of the things we want to prove is going to be easier. For example, let's rewrite this statement as its contrapositive. If p squared is even, then p is even. Let's identify the premises required to prove the conditional and its contrapositive, and also the conclusion, and which of these two seems easier to prove. So we have our antecedent, p squared is even, and our consequent, p is even. So the contrapositive, we switch and negate them. So as a contrapositive, if p is not even, then p squared is not even. Now let's take a look at our premises and our conclusion. The conditional itself has premise, p squared is even, and conclusion, p is even. The contrapositive has premise and conclusion, premise p is not even, and conclusion, p squared is not even. Now, algebraically, it seems easier to start with p and obtain p squared, and so the contrapositive might be easier to prove. Well, remember, definitions are the whole of mathematics. Let's try and prove that contrapositive. Now, a useful idea to keep in mind is don't use negative statements. Wait a minute, how about avoid negative statements? And the reason is that a positive statement tells you something, and a negative statement doesn't. Well, if p isn't even, then it's odd, and so we need a definition for what it means to be an odd number, and we might try something like, a number is odd if it's one more than a multiple of two. So let's try and prove our statement, and we'll prove our contrapositive. If p is odd, then p squared is odd. To prove a conditional, we can always start by assuming the antecedent. So suppose p is odd. And again, it's helpful to keep our destination in mind. We want to conclude that p squared is odd. So let's write down our conclusion, leaving some space to fill out the rest of the proof. Definitions are the whole of mathematics, all else is commentary. A number is odd if it's one more than a multiple of two, and our definition gives us two conditionals. If a number is odd, then it's one more than a multiple of two, and if a number is one more than a multiple of two, then it's odd. We have that p is odd, and so we know that p is 2k plus 1, and we get a conclusion like p squared is odd if we start with a number is one more than a multiple of two. So we know the preceding line should say that p squared is 2 times something plus one. So let's see if we can complete our bridge. We want to say something about p squared. We know what p is, so we know what p squared is. We can expand, and I can complete that last portion of the bridge by factoring two from these terms. So let's summarize our proof. Remember we can do that by rewriting it as a conditional if premises, then conclusion. So if p is odd, then p squared is odd. Of course, that's not what we were trying to prove, but that's okay. We can use the contra-positive. So contra-positively, if p squared is not odd even, then p is not odd even. How about proving that if p squared is odd, then p is odd. Now it seems like this should be pretty easy. We just prove that if p is odd, then p squared is odd, and this is the converse. But remember, the truth of the converse has no relation to the truth of the conditional. It requires a separate proof. So let's build that up. So again, it might be easier to work with the contra-positive, which allows us to work with p instead of p squared. And so our contra-positive is, now not odd is the same as even, so we can prove the contra-positive. So we can begin with our premise, set down our conclusion. Definitions are the whole of mathematics, all else is commentary. If p is even, then, and we can work our way towards conclusion. Join premise to conclusion and write our conditional. And again, that's not what we wanted to prove, but we can take the contra-positive and get what we want.