One to One
A function f from A to B is called one-to-one (or 1-1) if whenever f (a) = f (b) then a = b. No element of B is the image of more than one element in A.
In a one-to-one function, given any y there is only one x that can be paired with the given y.
f is one-to-one (injective) if f maps every element of A to a unique element in B. In other words no element of B are mapped to by two or more elements of A.
Onto
A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b. All elements in B are used.
f is onto (surjective)if every element of B is mapped to by some element of A. In other words, nothing is left out.
f is one-to-one onto (bijective) if it is both one-to-one and onto. In this case the map f is also called a one-to-one correspondence.
Notice that “f is one-to-one” is asserting uniqueness, while “f is onto” is asserting existence.
Let A and B be two finite sets such that there is a function f: A -> B. We claim the following theorems:
If f is one to one then |A| ≤ |B|.
If f is onto then |A| ≥ |B|.
If f is both one-to-one and onto then |A| = |B|.