One to One Onto and Bijective Functions

In this class, We discuss One to One Onto and Bijective Functions.

The reader should have prior knowledge of function basics. Click Here.

One to One or Injective function:

A mapping F: X -> Y is called one-to-one if distinct elements of X are mapped into distinct elements of Y.

Examples:

The diagram below shows the one-to-one function:

The first mapping is one-to-one because different elements in X have mapped to different elements in Y.

The second mapping is not one-to-one. Because elements 1 and 2 map to y1.

The function below is not one-to-one.

F: Z -> Z

F(x) = x^2 is not one to one.

Elements -3 and +3 both mapped to element 9.

Onto or Surjective Mapping:

A mapping F: X -> Y is onto if Range(F) = Y.

The below diagram shows the onto functions.

The first mapping is an onto function because the range of the function is = Y.

To understand better, every element in Y should have a mapping in the onto function.

The second mapping is not an onto function because the range is not equal to Y.

Bijective function:

A mapping is bijective if it satisfies both one-to-one and onto.