- 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?
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.