 Let's take a further look at transpositions and we'll introduce the following term so that we have something we can talk about. If I have some permutation in p of n, if it's a product of an even number of transpositions, we'll say that it has an even parity. And likewise, if it has the product of an odd number of transpositions, we're going to say that it has a odd parity. And one of the things we notice is that the parity of a permutation seems to be the same, even or odd, regardless of the actual transpositions that are used to produce it. And so we found enough evidence to suggest that we might want to try and prove the following theorem. The parity of a permutation is independent of the transpositions used to produce it. So let's talk about proof strategies. And so here's a potentially useful strategy, which is that we have this very, very general theorem. This is something that applies to all permutations in all permutation groups. And it's a very general result, which is why it's called a theorem. And we might have some difficulty trying to figure out how we can prove such a general result. So let's see if we can try and prove at least one case and see if that gives us any insight. And in this case, while we do want to prove something about all permutations, we might try to begin proving it by a single permutation. Now in general, proving one case isn't worthwhile because one example does not a theorem make. However, if we pick the right example, it may be useful. So let's try to prove this about the identity. And the reason that it is actually useful to prove this result for the identity is that we know that the identity is the element that is in every group of permutations. So while this is just a single permutation, at least it is something that's going to be true for all groups of permutations. So we know that the number of transpositions required to produce the identity is zero. Don't do anything. And that's an even number. And so we might try and prove that the identity has even parity. So let's think about that. Suppose I have the product of k transpositions that give us the identity. Now if I want to prove that the identity has even parity, then I know that I can write it as a product of an even number of transpositions. I want to show that I can't write it as a product of an odd number of transpositions. And what that means is I want to be able to show that I can't write the identity as a product of k plus 1 transpositions. And so that E is not some product of an odd number of transpositions. And let's think about that. What I'm starting with is I have information about k things. In particular, k transpositions give us the identity. So I do have knowledge, I have information about a set of k things. What I'd like to find is information about a set of k plus 1 things, namely that no product of k plus 1 transpositions gives us the identity. And that looks like an induction proof. I know something about k things, I want to know something about k plus 1 things. And so this suggests trying a proof by induction. So let's see how we can structure that. So we can break this proof into two parts. First of all, we want to prove that the identity can be written as the product of an even number of transpositions. And following that, we want to prove that the identity can't then be written as a product of an odd number of transpositions. If I can prove this, and I can prove that if I can write it as a product of k transpositions, I cannot write it as a product of k plus 1 transpositions, then that will give me this second result here. So the first part is fairly straightforward. You should be able to do that without any hints, and so I'm not going to give you any, but you should do it. The second part is going to follow from the first if we can prove that if I can write the identity as a product of k transpositions, I can't write it as the product of k plus 1 transpositions. So again, let's think about this as a strategy for proof. I want to prove that something can't be done. I want to prove that something is not possible. And frequently, when you want to prove something can't be done, you might try a proof by contradiction. Let's assume that we can write the identity as a product of k plus 1 transpositions. So I suppose I can write the product, the identity as a product of k plus 1 transpositions. And let's consider these last two transpositions. And we'll assume they're different because if they are the same, a transposition applied twice is just going to give you the identity. Nothing will happen. So if they are the same, I can just eliminate these two and have fewer. So let's assume that the last two transpositions are different with the last transposition being u, v help interchange u and v, and the one before it, well, there's only three possibilities. Either the preceding transposition can use u but not v. So it looks like u w. It keeps v but not u. So it looks like v w. Or maybe it has nothing in common with the last transposition. So it looks like w z or something like that. Now we'll only consider one case. You should be able to do the others. Suppose the last two transpositions are u w u v. Now we'll note that this product of transpositions can be rewritten as product u v v w. Again, you should be able to prove that that's the case. And the important observation here is u has been removed from that last transposition. It was in the last transposition but it's now been eliminated and it's been moved to the next to last transposition. Well, once we've done that, lather rinsed repeat, I can do the same thing with the second, next to last transpositions. u is in this last transposition here but I can move it to the one before. And in this way I can sweep all appearances of u into the first transposition only and that gives us the identity as a product of transpositions where u is at the first but in none of the other transpositions. Now we have a problem. In this product of transpositions, u is only going to be moved by the first. None of these other transpositions move u so u stays in place and then it's moved by this one but that can't possibly be the identity. And so we have our result that if I can write the product of the identity as a product of k transpositions, I cannot write it as a product of k plus one transpositions. And there's our proof. Well, actually there's a problem. The same logic applies when we write the identity as a product of k transpositions. In particular, if I could write the identity as a product of k transpositions ending this way, then I can sweep that u to the beginning and u is only moved by this and again it's impossible to write the identity as a product of k transpositions. There's a problem because this logic means I can't write the identity as a product of transpositions. We'll fix this problem in the next video.