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## Basic set operations

# 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.