 Okay, so let's have a look at these two examples here. I'm going to consider the binary operation This little circle On a set A, A being defined as the elements A, B, C, D, E. Don't see any ordering in that We're not talking about that. That's because it's alphabetical. Don't consider that. It's arbitrary set. Five elements Now for every ordered pair X and Y, so it can be A, C, C, A, D, C, B, E of this product set You've stuff we've learned before. We find this operation on that ordered pair by the following By this table. So we have to find it as a table. So if I want to find what This binary operation is, what do I go? Well A and A, well that gives me A. If I go B, C, let's go B, C So I've got to go B, C. I see that it's D If I go C, B, I go, let's go C, B That gives me D as well and B, C, so that gave me D. So I see that's happening See some commutative property coming in there But this is how we define this Binary operation. So you can see how much richer this is than just looking at functions Which is a very poor form. Well, of course we can do a lot with it What we use most of the time, but it does not give us this full picture of mathematics I can develop all kinds of these relations. So that's not addition. That's not multiplication. That's not Division. That's not subtraction It is something completely different than a function I can build something much richer and that is what we're going to take forward. Now example I might get back to, we'll have a look at the examples that we do get to, is this binary operation But if I take two, no, it's not part of this subset or this set A anymore, if I just take two elements and the binary operation of that is just going to be whatever I put first So if my set was the natural numbers If my set was the natural numbers, say that this is a binary relation on the set of natural numbers So if I take any pair of those and they are different, so if it's 301 and I do 1104 That is going to equal 301. So it's just whatever I put first is that And of course, you're now going to see this doesn't commute because by that binary operation 1004 that my binary operation with 301 is 1004 and of course here 1 is not 1 and 4 in other words This operation does not commute and that's what we're going to have a look at next