Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

48 0. Introduction and Preliminaries
or might not have repeats. The bijective functions are those that do not
have repeats and do not miss elements.
Image and Inverse Image
When discussing functions, we have notation for talking about an element
of the domain (say x) and its corresponding element in the codomain (we
write f (x), which is the image of x). Sometimes we will want to talk about
all the elements that are images of some subset of the domain. It would
also be nice to start with some element of the codomain (say y) and talk
about which element or elements (if any) from the domain it is the image
of. We could write “those x in the domain such that f (x) y,” but this is
a lot of writing. Here is some notation to make our lives easier.
To address the first situation, what we are after is a way to describe
the set of images of elements in some subset of the domain. Suppose
f : X → Y is a function and that A ⊆ X is some subset of the domain
(possibly all of it). We will use the notation f (A) to denote the image of A
under f , namely the set of elements in Y that are the image of elements
from A. That is, f (A) { f (a) ∈ Y : a ∈ A}.
We can do this in the other direction as well. We might ask which
elements of the domain get mapped to a particular set in the codomain. Let
f : X → Y be a function and suppose B ⊆ Y is a subset of the codomain.
Then we will write f −1 (B) for the inverse image of B under f , namely
the set of elements in X whose image are elements in B. In other words,
f −1 (B) {x ∈ X : f (x) ∈ B}.
Often we are interested in the element(s) whose image is a particular
element y of in the codomain. The notation above works: f −1 ({y}) is the
set of all elements in the domain that f sends to y. It makes sense to think
of this as a set: there might not be anything sent to y (if y is not in the
range), in which case f −1 ({y}) ∅. Or f might send multiple elements to
y (if f is not injective). As a notational convenience, we usually drop the
set braces around the y and write f −1 (y) instead for this set.
WARNING: f −1 (y) is not an inverse function! Inverse functions only
exist for bijections, but f −1 (y) is defined for any function f . The point:
f −1 (y) is a set, not an element of the domain. This is just sloppy notation
for f −1 ({y}). To help make this distinction, we would call f −1 (y) the
complete inverse image of y under f . It is not the image of y under f −1
(since the function f −1 might not exist).