# Walk Through of RSA Public-key Encryption for Beginners - Part III:Bijection Functions

Today we wil be going over bijection functions. We will proceed to understand:

1. What is a function?
2. What is onto function?
3. What is one-to-one function?
4. What is bijection function?
5. What make a function bijection?

6. What is a function?

A function can be viewed a mapping from elements in set A to elements in set B. For input x in A (domain), output f(x) must be in B (range). We can also denote such function f as f: A—>B

2.What is onto function

An onto function is a function that for everyone output f(x) in B, there is an pre-image of it in A. In other words, all elements in B is linked to some element in A.

See the image below; F is not onto where as G is.

3. What is one-to-one function

A one-to-one function is a function that  for all a, a' in A, if f(a)= f(a') , then a=a'.

In other words, elements in A cannot share the an element in B.

See the image below; F is one-to-one while G is not.

1. What is a bijection function.

A bijection fuction is both onto and one-to-one.

5.What make a function bijection?

Lemma: If a function f maps set A–>A, and it has an inverse function g(x) such that g(f(x))=x, then f is a bijection function.

We will leave the proof of the above Lemma as an exercise.