relation

in mathematics, a relation is how sets of numbers relate to each other

a relation consists of a relation name, typically a letter, and an equation
for example: $f:f(a,b)=a^2-b^3$

types of relations

set $X$ is the input set of the relation and set $Y$ is the output set of the relation

there are four properties that a relation may have:

1. each element of X must be paired with at least one element of Y
2. no element of X may be paired with more than one element of Y
3. each element of Y must be paired with at least one element of X
4. no element of Y may be paired with more than one element of X

one to one (also called a bijective relation) - has properties 1 2 3 4
each distinct input corresponds to a distinct output, and vice versa. this makes the relation invertible
for example, $f(x)=3x+1$

many to one - has properties 1 2 3
an output can correspond to many possible inputs
for example, $f(x)=x^2$

one to many - has properties 1 3 4
an input may correspond to many possible outputs
for example, $x=f(x{)}^2$

many to many - has properties 1 3
an input can correspond to many outputs and an output can correspond to many inputs.
for example, $f(x,y):x^2+y^2=1$

functions

a function is a relation in which each input has exactly one output - a one-to-one relation or a many-to-one relation