 So from time to time you'll be asked to prove something in mathematics. So let's talk a little bit about proof in mathematics. So, a mathematical proof is a sequence of logically justified statements. Now, that really doesn't tell us anything, but the important idea here is that because it's a sequence, it has a last line, and that last line is the conclusion of the proof. This is actually fairly important to keep in mind, because it means that the last thing in your proof is what you have actually proven, and that's something that you'll want to be careful with. It's a fairly involved subject, and in general, this is going to be a very, very, very, very, very short overview for purposes of what we'll be using it for. So, one of the things that we'll be proving are conditional statements, and these are actually fairly common. They are statements of the form, if something, then something. For example, if k is an even number, then k squared is even, or if p is prime, then p is equal to seven, or if n is composite, then n can be written as the product of two prime numbers. Now, I make no claims that any of these statements are actually true, but they are here to illustrate some key parts. If I have a conditional statement, the if portion of the conditional is called the antecedent. So, here, for example, I have if k is an even number, that's our antecedent. p is prime, that's our antecedent. n is composite, that is the antecedent. And so, the if portion of the conditional is the antecedent. The then part is called the consequent. So, here, k squared is even, p is seven, and can be written as the product of two prime numbers. Those are all the consequence of the conditional. And looking at a conditional in this fashion helps us to construct a fairly standard process for proving any conditional statement. So, suppose I have a conditional statement, if a, then b. So, if I want to prove this statement in general, we may always begin by assuming that the antecedent is true. So, I know that a, whatever it is, is true. And, again, proof is a sequence of logically connected statements. So, from a, I might conclude that c is true, or d, e, f, and g are true, a bunch of other things are true. And at some point, I'm going to have drawn enough conclusions that b is going to emerge. And so, I have my sequence of statements. I'm starting out with a, my antecedent, and I'm drawing some other conclusions and eventually ending with b. And at this point, I'm done because b, the last line, the last thing I say in the proof, is our conclusion, which is what we want for our proof. However, it's fairly good form to reiterate our conclusion in the form if first line, then last line. So, if a, then b, and that is what I have in fact proven. Well, let's take a look at an actual example here. So, let's take a look at the example if k is an even number, then k squared is even. So, it's often convenient, again, to identify the antecedent inconsequent because that tells us where we start, that's our antecedent, and where we want to end up, that's our consequence. So, the antecedent, again, is the if part of the conditional k is an even number. And again, we may always assume that the antecedent is true. So, k is an even number. And because mathematicians don't like to compact their sentences too much, I'll say let k be an even number. Now, to build the rest of the proof, we need to draw logical conclusions based on the earlier statements. And an extremely useful tactic is to rely on the definitions. And part of the reason for that is the definitions give us a built-in framework of what we can say without really having to go into too much justification for it. So, k is an even number. Well, what does that mean? By definition, even numbers are products of 2 and something else, so that tells me k is 2 times something. And keep in mind that where I want to end up, I'd like to say something about k squared. And because of that, I can use a little bit of algebra. There's k squared. And, well, I don't quite know what I want to do, but I do know that I want to end up with the statement k squared is even. So another useful tactic in building a proof is to work backwards from the end. Now, we have to be a little bit careful with this. Remember that the conclusion is the last line of the proof. The last thing I want to say, k squared is even. So that's going to be at the very end of the proof. And what I'm going to do is I'm going to write it down here. And that leaves me a little bit of space to write some intervening things. I am not going to write anything after this. If I write anything after this, I will not have proven k squared is even. I will have proven whatever it is I write afterwards. So I'll leave some space. And, again, here's why definitions are useful. They work in both directions. So forward direction, k is even, so I know k is 2p. Backward direction, I know k squared is even. I can take a step back and I know that k squared is a product of 2 and something else. And what I have to do is I have to bridge the gap between where I was here and where I want to be here. And, again, I can do a little bit of algebra. k squared is 2 times 2p squared. And so that tells me that k squared is a product of 2 and something else and k squared is even. And we'll do a quick check just to make sure our proof flows in the direction it should. I have to start here. I have to be able to end here. k is even, product of 2 and something else, k is 2p, k squared is 4p squared, do a little bit of algebra, k squared is the product of 2 and something else, k squared is even, and my proof is good.