- 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.