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Video transcript
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.