Mathematical closure

- So I have a very simple
equation written here

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

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

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

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

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

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

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

B is an integer.

If we're adding these two integers,

are we definitely going
to get another integer?

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

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

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

come up with some cases.

Well, what if a and b are both one,

then for sure, c is going to be two.

And c is going to be an integer.

Well is that always going to be the case?

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

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

a situation like this, when you're able

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

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

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

a or b again or it could
be another distinct member

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

of the set, whenever you have this,

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

So when you're performing
the addition operation

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

C is also going to be a member of that set

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

we're going like this and we are

performing this addition operation.

Whenever you see a circumstance like this,

we call, we say that this set,

the set of integers is
closed under addition.

Let me write that down.

So this means that the
set of integers, integers

is closed under, under addition.

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

of closure, it's the general term

that we're talking about is closure

when you take some, especially in higher

mathematics course but
you might even hear it

some of your high school courses

and it might seem like
this really advanced thing

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

you have some operation
that you can perform

on members of the set and if every time

you perform that operation
on members of the set

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

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

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

the operation is addition, you say

the set is closed under addition
and we can ask ourselves

that same question about other things.

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

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

Is it closed under
multiplication, multi-plication.

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

And then think about, is
it closed under division?

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

Is it closed under subtraction?

I encourage you to think about

all these different operations.

Let's take it one by one. Multiplication.

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

you actually always
will get another integer

so the set of integers is
closed under multiplication.

Now, what about division?

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

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

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

One divided by two. One
and two are both integers

but what's that going to give me?

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

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

to extend our sets right over here.

All integers are also rational but

all rationals are not integers necessarily

so this is rational,
rational numbers here.

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

and this is two when
we performed division,

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

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

but it is a rational number
so the set of integers

is not closed under
division. Subtraction? Sure!

And once again, I'm
not going to rigorously

prove it here but take any two integers,

you indeed are going to get another,

you are going to get another integer.

So the whole idea here is to just give you

a brief introduction to
the notion of closure.

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

an operation on members of that set

and you always get a
member of your original set

then that set is closed
under that operation.

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

in this case, we talked
a lot about integers

and we talked about
addition, multiplication,

division, subtraction but you could think

about the set of polynomials
and whether they are

closed under addition,
which they actually are

or the set of rational numbers.

Are they closed under exponentiation?

There's all sorts of
things we can think about

but hopefully, this gives you a good sense

of what we mean when we say closure

or for a set being closed
under some operation.