catalogue

If you have two sets A and B a function f from A to B assigns one member of A a corresponding element in B. If x is an element in A and y is an element in B you can write y = f(x) and y is called the image of x under the function f. you can also write

f : A→B

A is called the domain of f. The subset of B which contains all of the images of elements of A is called the range.

Flavors of functoins

Injective

An injective function is one where each element of B is the image of  no more than one element of A

In other words if y = f(x1) = f(x2) x1 = x2

Surjective

A surjective function is one where every element of B is the image of at least one element of a. In other words this is saying B is the range of A

 Bijective

 If a function is both injective and surjective it is called bijective. A bijective function is said to have a one-to-one correspondence between the mebmers of A and B

Composite Functions

same idea from elementary algebra only there are some important things to note:

given two functions f and g

if f and g are injective then g◌f is injective

if f and g are surjective then g◌f is surjective

if f and g are bijective g◌f is bijective

Inverse Functions

A function f : A→B has an inverse if and only if it is bijective. Furthermore the inverse of it is a bijective function from B to A