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- So I have a very simple
equation written here

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a plus b is equal to c and
what I want to think about

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if we know that a is an
integer, so let's say that this

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is the set of all integers
right over here. Integers.

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We know that a is an integer
so let's say that's a there.

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We also know that b is an
integer, could be the same integer

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but I'll draw it so it
looks like it's another one,

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it doesn't have to be but they
could be the same integer.

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B is an integer.

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If we're adding these two integers,

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are we definitely going
to get another integer?

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Do we know that c is
going to be an integer?

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And I encourage you to pause
the video and think about it.

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And when you've thought
about it, you might have

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come up with some cases.

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Well, what if a and b are both one,

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then for sure, c is going to be two.

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And c is going to be an integer.

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Well is that always going to be the case?

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Well, it is. There are ways to prove it.

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I'm not going to go into
it now but when you have

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a situation like this, when you're able

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to take two members of
a set, in this case,

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two members of a set and
perform an operation on it,

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and always get another member
of the set and it could be

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a or b again or it could
be another distinct member

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of the set, I'll draw it
as another distinct member

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of the set, whenever you have this,

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so you have some operation,
in this case, it is addition.

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So when you're performing
the addition operation

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on two members of the set,
you get, you stay in the set.

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C is also going to be a member of that set

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so one way to think about it, we're going,

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we're going like this and we are

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performing this addition operation.

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Whenever you see a circumstance like this,

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we call, we say that this set,

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the set of integers is
closed under addition.

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Let me write that down.

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So this means that the
set of integers, integers

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is closed under, under addition.

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So once again, you're
going to hear this idea

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of closure, it's the general term

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that we're talking about is closure

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when you take some, especially in higher

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mathematics course but
you might even hear it

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some of your high school courses

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and it might seem like
this really advanced thing

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but all it's saying is you have a set,

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you have some operation
that you can perform

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on members of the set and if every time

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you perform that operation
on members of the set

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you get, you stay in
the set, the result is

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a member of the set, then you
say that the set is closed

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under that operation, the
set is closed, in this case,

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the operation is addition, you say

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the set is closed under addition
and we can ask ourselves

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that same question about other things.

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Is the set of, is the set
of integers closed under,

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closed under, actually,
let's take some examples.

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Is it closed under
multiplication, multi-plication.

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And I encourage you to pause
the video and think about it.

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And then think about, is
it closed under division?

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Is it closed under division?
Is it closed under subtraction?

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Is it closed under subtraction?

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I encourage you to think about

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all these different operations.

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Let's take it one by one. Multiplication.

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If you take two integers
and you were to multiply it,

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you actually always
will get another integer

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so the set of integers is
closed under multiplication.

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Now, what about division?

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Well, it's very easy to
prove a case, in which,

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I'm dividing two integers and
not getting another integer.

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For example, I could say, let's
just do a very simple one.

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One divided by two. One
and two are both integers

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but what's that going to give me?

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That's going to give me 1/2
which is not, another integer.

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This is a rational number but
not an integer so if we were

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to extend our sets right over here.

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All integers are also rational but

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all rationals are not integers necessarily

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so this is rational,
rational numbers here.

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This case of one divided
by two so that's one

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and this is two when
we performed division,

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we went out of the set, so
when we performed division,

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we out of that set to
something that's not an integer

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but it is a rational number
so the set of integers

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is not closed under
division. Subtraction? Sure!

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And once again, I'm
not going to rigorously

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prove it here but take any two integers,

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you indeed are going to get another,

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you are going to get another integer.

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So the whole idea here is to just give you

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a brief introduction to
the notion of closure.

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The idea that if you have
a set and if you perform

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an operation on members of that set

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and you always get a
member of your original set

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then that set is closed
under that operation.

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What matters is that the set
that you're talking about

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in this case, we talked
a lot about integers

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and we talked about
addition, multiplication,

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division, subtraction but you could think

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about the set of polynomials
and whether they are

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closed under addition,
which they actually are

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or the set of rational numbers.

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Are they closed under exponentiation?

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There's all sorts of
things we can think about

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but hopefully, this gives you a good sense

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of what we mean when we say closure

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or for a set being closed
under some operation.