Ex. Suppose we have the following sets:

A = { 0, 1, 2, 3, 4, 5 }

B = { 0 , 1, 1/2, 1/3, 1/4, 1/5 , 4, 9, 16, 25 }

And we allow the function f to be as follows:

f: A B = f(x) = 1 / x.

g: A B = g(x) = x*x.

g: A B = { f(0) = 25, f(1) = 1/3, f(2) = 1/5, f(3) = 1, f(4) = 1}

Notice that all of the domain and codomain do not have to be used. Also, elements in the codomain can be used multiple times, however, an element in the domain can only be mapped once.

Definition. The image is a subset of the domain that is used as actual inputs. The range is subset of the codomain that is used by the image. The preimage is the subset of the domain that contains all the elements which could cause a given range. The preimage is denoted as f ^-1, and while it appears to be the inverse function, it is not the same thing.

The preimage is formally defined as:

f ^-1 (B) = { x A : f(x) B }.

The preimage consists of all the elements of A that, once passed through the function, become an element passed to the preimage. I know that sounds really confusing, but after an example it'll be pretty clear how simple it is.

Ex. Using our above function f and applying the image { 1, 2, 3, 4, 5 }, the range is { 1, 1/2, 1/3, 1/4, 1/5 }.

Ex. Suppose we have the following sets:

A = { -2, -1, 0, 1, 2, 3, 4, 5 }

B = { -1, 0 , 1, 2, 3, 4, 5 }

And the following function:

f: A B = f(x) = Abs) = (the absolute value of x)

f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 2, f(3) = 3, f(4) = 4, f(5) = 5

When using the image { -1, 0, 1 }, the range  is { 0, 1}.

The preimage of { 1 } is { -1, 1 }.

The preimage of { 1, 2 } is { -2, -1, 1, 2 }.

The preimage of { -1 } is the null set (empty set), since no values in A could give a function output of { -1 }.

Definition. A function is onto and called a surjection if every element in the domain maps to every element in the codomain. A function is one-to-one and called an injection if no elements in the range are used more than once.

Ex. Given A = { 1, 2, 3, 4 } and B = { 0 } and f: A B, and f(1) = 0, f(2) = 0, f(3) = 0, and f(4) = 0, f is onto and therefore a surjection

Given f: B A, and f(0) = 3, f is one-to-one and therefore an injection.

Definition. A function is a bijection if it both one-to-one and onto. Two sets have the same cardinality if and only if a bijection can be formed between the two sets.

Ex. A bijection can be formed between the integers , the natural numbers , and the rational numbers .