 Now, before we move on to a totally new concept, just filling up our toolbox with some knowledge, we have looked at this distributive property of our operations. And so in this video, I just want to bring down the concept that we can really combine these. So imagine that we have two binary operations, as per usual, we'll have these two. And we'll just talk about this distributive properties of combining these two. In the first bit, we're going to call left distributive. Left distributive. And let's have the following example. I think I've got an example here. I'm going to say A and then I'm going to have B, for instance, in C. B and C, there we go. And I'm going to say that combining these two are left distributive if we really have the following properties. Then if I were to do this, it is going to equal the following that I have this. And then A with C. If this property holds, if I were to do this, I'd call it left distributive. And if I were to do the following, right distributive. Right distributive. And it will say the following. And in this instance, I'm going to have, let's say, again my B and C and then A. And if that equals, if this were to equal, there's my A. If this were to equal the B and the A. And we have the C and the A. If those hold, we're going to say that we have these properties. This is a distributive or at least right distributive property of this. Let's just be clear about this, of this binary operation with respect to that one. And you might wonder, well, it's natural for us just to see this in terms of plus and addition and multiplication. And of course, these will hold. But what if we were to do something a bit different? We said that we are expanding our world so we are not only seeing these as addition and multiplication. We might see it as something different. And I've got a nice example here. So let's have these elements X and Y and they're all elements of the integers. Any of these elements, any two elements of the integers, I should say. And I have the following. I would say that I have a property that my binary operation, that one is addition. I define that one as addition, but I define this one as the following. That is X squared times Y. So whenever I see that, it is X squared plus Y. So let's have a look at this left distributive property. So I'm going to start with the following. I'm going to say A and then B and Z. So that's what I have there. So is it left distributive? So remember there is my left distributive. So it's going to be A with B and A with C. And remember this is now, this is defined as that. I'm going to have A squared B and I'm going to have, that is addition plus I'm going to have A squared C. So that is what I would have if I were to do this. But remember, just look at this. This one says square the first one, multiplied by the second one. So it's the same as doing the following. This would be the same as that because there is my defined binary operation. So it's going to be A squared times B plus C. And that's A squared B plus A squared C. And those two are exactly the same. So we would say that this is left distributive. Is it right distributive though? Let's just have a look at that. Let's just have a look at that. If I were to have B and C and then A. So it's right distributive with, my example is with respect to this. With respect to this. So if I were to do this normal distribution, I would really get out here B, A. And I would get C and A. That is what I would get. This normal distribution, normal distributive property. Now this is now that which is going to give me B squared A plus. On this side I'm going to have, that's my plus. That's C squared times A. So B squared A plus C squared A. That is what I would get. But if I just had to look at this. Let's see if this is the same. So it's this one squared times that one. That's what I have there. So I'm going to have B, that's a plus. B plus C squared times A. And what am I getting to get here? I'm going to get B squared plus 2BC plus C squared all multiplied by A. So I'm going to get AB squared plus 2 ABC plus AC squared. And these two are definitely not the same. They are definitely not the same. So this is not right distributive with respect to that. This binary operation is not left distributive with respect to that. Because if we were to do this we get two different answers. If we do it in the right distributive, the right distributive on that side, we see, now I mustn't get these two. Please don't get these two confuses. I've just said this is left distribution. Is it left distributive? And this is, is it right distributive? This with respect to that. And yes it is, and indeed no it is not.