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## Statistics and probability

# Intersection and union of sets

To find the intersection of two or more sets, you look for elements that are contained in all of the sets.
To find the union of two or more sets, you combine all the elements from each set together, making sure to remove any duplicates. Created by Sal Khan.

## Video transcript

What I want to do in this
video is familiarize ourselves with the notion
of a set and also perform some operations on sets. So a set is really just a
collection of distinct objects. So for example, I could have
a set-- let's call this set X. And I'll deal with
numbers right now. But a set could
contain anything. It could contain colors. It could contain people. It could contain other sets. It could contain cars. It could contain farm animals. But the numbers will be
easy to deal with just because-- well, they're numbers. So let's say I have
a set X, and it has the distinct objects in it,
the number 3, the number 12, the number 5, and the number 13. That right there is a set. I could have another set. Let's call that set Y.
I didn't have to call it Y. I could have called it A.
I could have called it Sal. I could have called it a
bunch of different things. But I'll just call it Y. And let's say that set
Y-- it's a collection of the distinct objects, the
number 14, the number 15, the number 6, and the number 3. So fair enough, those are
just two set definitions. The way that we typically
do it in mathematics is we put these
little curly brackets around the objects that
are separated by commas. Now let's do some basic
operations on sets. And the first operation that I
will do is called intersection. And so we would
say X intersect-- the intersection of X
and Y-- X intersect Y. And the way that I
think about this, this is going to
yield another set that contains the elements
that are in both X and Y. So I often view this
intersection symbol right here as "and." So all of the things that
are in X and in Y. So what are those things going to be? Well, let's look at
both sets X and Y. So the number 3 is in set
X. Is it in set Y as well? Well, sure. It's in both. So it will be in the
intersection of X and Y. Now, the number 12, that's
in set X but it isn't at Y. So we're not going
to include that. The number 5, it's
in X, but it's not in Y. And then we have
the number 13 is in X, but it's not in Y. And so over
here, the intersection of X and Y, is the set that
only has one object in it. It only has the number
3 So we are done. The intersection
of X and Y is 3. Now, another common
operation on sets is union. So you could have
the union of X and Y. And the union I often view--
or people often view-- as "or." So we're thinking about
all of the elements that are in X or Y. So
in some ways you can kind of imagine that
we're bringing these two sets together. So this is going to
be-- and the key here is that we care-- a set is a
collection of distinct objects. And the way we're
conceptualizing things right here, this
is the number 3. This isn't like
somebody's score on a test or the number of
apples they have. So there you could
have multiple people with the same number of apples. Here we're talking about
the object, the number 3, so we can only have a 3 once. But a 3 is in X or Y,
so I'll put a 3 there. A 12 is in X or Y. A 5 is in
X or Y. The 13 is in X or Y. And just to simplify
things, we really don't care about order if
we're just talking about a set. I've just put all of the
things that are in set X here. And now let's see what we
have to add from set Y. So we haven't put a 14 yet. So let's put a 14. We haven't put a 15 yet. We haven't put the 6 yet. And we already have
a 3 in our set. So there you go. You have the union of X and Y. And one way to visualize sets
and visualize intersections and unions and more
complicated things, is using a Venn diagram. So let's say this whole box
is-- you could view that as the set of all numbers. So that's all the
numbers right over there. We have set X-- I'll just draw
as circle right over here. And I could even draw
the elements of set X. So you have 3 and
5 and 12 and 13. And then we can draw
set Y. And notice, I drew a little overlapping
here because they overlap at 3. 3 is an element in
both set X and set Y. But set Y also has the
numbers 14, 15, and 6. And so when we're talking
about X intersect Y, we're talking about where
the two sets overlap. So we're talking about this
region right over here. And the only place that they
overlap the way I've drawn it is at the number 3. So this is X
intersect Y. And then X union Y is the combination
of these two sets. So X union Y is
literally everything right here that
we are combining. Let's do one more
example, just so that we make sure we understand
intersection and union. So let's say that I have set
A. And set A has the numbers 11, 4, 12, and 7 in it. And I have set B, and it has
the numbers 13, 4, 12, 10, and 3 in it. So first of all,
let's think about what A-- let me do that in A's color. Let's think about
what A intersect B is going to be equal to. Well, it's the things
that are in both sets. So I have 11 here. I don't have an 11 there. So that doesn't make
the intersection. I have a 4 here. I also have a 4 here. So 4 is in A and B. It's in A
and B. So I'll put a 4 here. The number 12, it's in A and
B. So I'll put a 12 here. The number 7 is only in A. And the number, I
guess, 13, 10 and 3 is only in B, so we're done. The set of 4 and 12 is the
intersection of sets A and B. And we could even,
if we want to, we could even label
this as a new set. We could say set C is the
intersection of A and B, and it's this set
right over here. Now let's think about union. Let's think about A-- I
want to do that in orange. Let's think about
A union B. What are all the elements
that are in A or B? Well, we can just literally put
all the elements in A, 11, 4, 12, 7. And then put the things in
B that aren't already in A. So let's see, 13. We already put the 4 and
the 12, a 10 and a 3. And I could write this
in any order I want. We don't care about order if
we're thinking about a set. So this right here is the union.