Let's add some transformation that maps elements in
set X to set Y.
We know that we call X the domain of T.
So that's my set X and then my set that I'm mapping into, set
Y, that's the codomain.
We know that T is a transformation that if you
take any member of X and you transform it, you'll associate
it with a member of set Y.
You'll map it to a member of set Y.
That's what the transformation or the function does.
Now, if we have some subset of T, let's call A to be some
subset of T.
So let me draw A like that.
This notation right here just means subset,
some subset of T.
We've defined the notion of an image of T of A like that,
which is the image of A, of our subset A, under T.
We've defined this to be equal to the set-- let me write it
here-- the set of all-- where if we take each of the members
of our subset, it's the set of all of their transformations.
And, of course, these are going to be some subset of Y
right there.
So we essentially take each of the members of A.
This was one of them.
You find its transformation.
It's that point.
You take another member of A.
This is all set A right here.
Take another member of A.
Find its transformation.
Maybe it's that point.
You keep doing that.
Find it's transformation.
Maybe it's that point.
And then the set of all of those transformations, maybe
it's this blob right here, we call this the
image of A under T.
Now, what if we wanted to think about
the opposite problem?
What if we were to start with set Y, which is our codomain,
So that's Y, and we were to have some subset of Y.
Let's call our subset of Y S.
So S is a subset of our codomain Y.
And I'm curious about what subset of X maps into S.
So I'm curious about this set.
I'm curious about the set of all vectors that are members
of my domain such that they're mapping or the transformation
of those vectors ends up in my subset of S.
So what I'm saying is, look, if I take my domain, there
must be some subset of vectors right here, where if I take
any member of this set, it will map into these guys, and
that's what I'm defining right here.
This is equal to that guy.
So I'm literally saying what are all of the members of X
where those members of X all map into S?
Now, I want to make a very subtle nuance here to point
out something here.
I'm not saying that every point in S
necessarily gets mapped to.
For example, maybe there's some element in S right there
that no element in X ever gets mapped to from our
transformation T.
That's OK.
All I'm saying is that everything in this set maps to
something within S right here.
And what we call this set right here, the notation is
the inverse T of S, but this is equal to the
pre-image of S under T.
So this is S.
This is the pre-image of S under T.
And that makes sense.
The image, we go from a subset of our domain to a subset of
our codomain.
Preimage, we go from a subset of our codomain, and we say
what subset of our domain maps into that
subset of our codomain?
Now let me ask you an interesting question, and this
is kind of for bonus points.
What is the image of our pre-image under S?
So if we take this guy, this is essentially the image of
this guy right here, right?
This part right here is the pre-image of S right here.
Now, if we take the image of this, we're saying if we take
every member of this, what vectors do they map into?
All of them are going to be within S, so they're going to
map within S, but they don't necessarily map to
everything within S.
So this is going to be some subset of S.
So this right here is going to be some subset of
our original S.
It's not necessarily equal to S, but it's a subset of it.
And so this is I think the motivation for where the
notation comes from.
We can construct a subset of S by taking the image of the
pre-image of S.
We can kind of view the image and the pre-image kind of
canceling out, and that's why the inverse notation was
probably introduced.
Now this is all very abstract.
What I'm going to do in the next video is actually
calculate or determine the pre-image of some
subset of my codomain.