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Subset, strict subset, and superset

Video transcript
Let's define ourselves some sets. So let's say the set A is composed of the numbers 1. 3. 5, 7, and 18. Let's say that the set B-- let me do this in a different color-- let's say that the set B is composed of 1, 7, and 18. And let's say that the set C is composed of 18, 7, 1, and 19. Now what I want to start thinking about in this video is the notion of a subset. So the first question is, is B a subset of A? And there you might say, well, what does subset mean? Well, you're a subset if every member of your set is also a member of the other set. So we actually can write that B is a subset-- and this is a notation right over here, this is a subset-- B is a subset of A. B is a subset. So let me write that down. B is subset of A. Every element in B is a member of A. Now we can go even further. We can say that B is a strict subset of A, because B is a subset of A, but it does not equal A, which means that there are things in A that are not in B. So we could even go further and we could say that B is a strict or sometimes said a proper subset of A. And the way you do that is, you could almost imagine that this is kind of a less than or equal sign, and then you kind of cross out this equal part of the less than or equal sign. So this means a strict subset, which means everything that is in B is a member A, but everything that's in A is not a member of B. So let me write this. This is B. B is a strict or proper subset. So, for example, we can write that A is a subset of A. In fact, every set is a subset of itself, because every one of its members is a member of A. We cannot write that A is a strict subset of A. This right over here is false. So let's give ourselves a little bit more practice. Can we write that B is a subset of C? Well, let's see. C contains a 1, it contains a 7, it contains an 18. So every member of B is indeed a member C. So this right over here is true. Now, can we write that C is a subset? Can we write that C is a subset of A? Can we write C is a subset of A? Let's see. Every element of C needs to be in A. So A has an 18, it has a 7, it has a 1. But it does not have a 19. So once again, this right over here is false. Now we could have also added-- we could write B is a subset of C. Or we could even write that B is a strict subset of C. Now, we could also reverse the way we write this. And then we're really just talking about supersets. So we could reverse this notation, and we could say that A is a superset of B, and this is just another way of saying that B is a subset of A. But the way you could think about this is, A contains every element that is in B. And it might contain more. It might contain exactly every element. So you can kind of view this as you kind of have the equals symbol there. If you were to view this as greater than or equal. They're note quite exactly the same thing. But we know already that we could also write that A is a strict superset of B, which means that A contains everything B has and then some. A is not equivalent to B. So hopefully this familiarizes you with the notions of subsets and supersets and strict subsets.