 In the previous video, we learned about binary operations related to the idea of a binary operations, the idea of closure. So let's imagine we have an operation on some set x. So what that means, of course, is that circle here is an operation of x cross x to x. And in case you're wondering, you can draw this circle in latex by doing backslash C-I-R-C, short for circle. So we have a binary operation on the set x with respect to the circle operation. What if we have some subset y inside of x? Can we define a binary operation on y using the binary operation that's on x? Well, is there a way of defining circles so that y cross y gives us something? Well, since y belongs to x, then because we have an operation for every possible pairing of x and x, if we restrict the domain to only be operands coming from y, then of course this result here will be well-defined. But in order to guarantee that this itself is an operation, we need that when you combine things from y to y, you always get something in y. Is that always the case? We know that it'll always go to x, but can we restrict it just to y? And if we can, we say that the subset is closed with respect to the operation. That is, the resultant of combining any two elements from y gives you something in y. And again, this is a phenomenon we see a lot. Like on the previous video, we had talked about the following important sets. N, z, q, r, and c. We talked about with respect to addition and multiplication that these were all binary operations on these sets. And in fact, what we can say a little bit better is that r is actually closed with respect to addition. The same addition of complex numbers. The addition of real numbers is just a special case of addition of complex numbers because a real plus a real always gives you a real. And likewise, addition of two rational numbers is just a special case of real addition. The sum of two rationales will always give you a rational. And the same thing is true for integers. The sum of two integers is really just a special case of rational addition. And same thing with natural numbers as well. It's just a special case of integer addition. Same thing with multiplication. Now, you might have noticed in the previous video that when we talked about binary operations, I did not mention subtraction or division. And there are some, there's some reasons to mention there. So first of all, when it comes to subtraction, subtraction, as our usual sense, and its usual sense, subtraction is an operation on c. It's an operation on r. And in fact, the subtraction operation on the real numbers is just a restriction of complex subtraction. A real minus a real is, in fact, a real again. Same thing is true for rationales and same thing is true for integers. I mentioned, though, that subtraction is not, it's not an operation. It's not an operation on the natural numbers. Because subtraction, if you subtract two natural numbers, you might not get a natural number back. So, you know, case in point, we could take something like zero minus two. That gives us negative two, which is not a natural number. And so subtraction, although subtraction is an operation on the integers, if you subtract any two integers, you get back an integer. If you take the subset of the natural numbers, the difference of two natural numbers is not necessarily natural number here. And so we would say that n is not, it's not closed. It's not closed under subtraction, viewing it as a subset, right, of the integers. But on the other hand, the integers, the integers are closed under subtraction. Again, viewing this as a subset of, like, say the rational numbers or something. Or we could also say that the natural numbers are closed under addition with respect to integers. So there is an important distinction that one has to make there, that the closure principle. If you restrict the operation to a smaller subset, will the resultants always fall inside of the smaller set? And that's not always the case. Like we see here with the natural numbers. What about division? When it comes to division, division is not an operation either for some of the same reasons, right? Like if you take, for example, one divided by two, this is not an integer. This is not an integer, even though one and two are integers, right? So you don't have closure under addition, under division, like if you do integers. But you also have problems with even the rationals, right? If you take, like, say two divided by zero, this is not inside the rational field, even though two and zero are inside that set. So you have some issues there. The problem, the problem here isn't necessarily closure. The problem is division in its usual sense is not an operation because division would be, it would be a function from q to cross, well, you basically get q take away zero because we can't divide by zero and that then produces something in q, right? So the fact that the sets don't exactly match up with each other exactly when it comes to division. Now there is one way of sort of fixing that using subsets that's related to this idea of closure. When it comes to division, right, we usually think of the following type sets. So we're going to take the set q star. This is going to be the set right here. This is going to be the set of rational numbers. So we're going to take X inside of the rationals, but X is not zero. So we just take the nonzero rationals. This is often defined to be q star. We do a similar thing for our star and we do a similar thing for C star right here. These sets represent the nonzero rationals, reels and complex numbers. I should mention that q star, of course, is obviously a subset of q and this set right here is closed. It's closed under multiplication. Q star is closed under multiplication. Likewise, our star is closed under multiplication and C star. C star is likewise closed under multiplication. Of course, rational, real and complex multiplication in all those situations. And so those are important subsets. Another reason why these subsets are important is because in all of these cases we can define division. Division, we can take as a map from q star cross q star. And we map it onto q star. And so we can't necessarily define division in all situations, but we can actually take the quotient on this subset that's closed under multiplication. I should mention though that q star, r star and C star, these sets are not. They're not closed with respect to addition. That if you take the sum of two nonzero rationals, you can back to zero. For example, one half plus negative one half is equal to zero. And so you get some issues there, right? And so you can see why we kind of play around with sets at times. Sometimes I want to focus on this set because it's closed under this operation, maybe not this other one. Sometimes I want to focus on this set because this operation like division is defined even though addition no longer becomes an operation. The sets for which form the set in play matters, right? Because we need the operands and the resultant to be part of all of those sets like so. I want to mention that when it comes to division though, like if you try to define something like nonzero integers, right? Division still doesn't work in that situation like we saw before. One half is not an integer. It's not even a nonzero integer. And so we don't necessarily focus on the nonzero integers as often as we do say q or c because we can make multiplication into closed subset. And we get multiple we also use division. I should mention though that if you do take like nonzero integers, you would be closed under additions close under multiplication. Excuse me. You're not close under addition. You need to close your under multiplication, but division is not extendable here. And so therefore we don't really talk about z star too much. Probably one potentially could do it. Let's see another example we saw in the previous video talking about function composition. Just as a reminder, if you take x to the x, right, this is the set of functions of the form f goes from x to x. Right, this is this is a this is a set with a binary operation of function composition. Right, so we have this binary operation you have x to the x comma circled. In this case here circle is genuine the real McCoy function composition and important subset of this thing is s sub x. Remember x sub x this right here is the set of permutations set of permutations from x to x. That is the function x to x. These functions f do not have to be bijective functions in sx. They are bijective. They have inverses function inverses here. This is an important this is an important subset of x to the x. And in fact, this subset is closed under function composition x sub x here is closed under composition. In which case it's not often called function composition. In that case, it's often called permutation multiplication, which might seem a little bit weird. But that's what we call we call it multiplication of the permutations. It's a very important subset, which is closed under function composition.