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Function - One to One & Onto

 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|.

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