Changes

|}

|}

For example, or more precisely, contra example, the function ''f'' : ''X'' → ''Y'' that is depicted below is neither total at ''Y'' nor tubular at ''Y'', and so it cannot enjoy any of the properties of being sur-, or in-, or bi-jective.

For example, the function <math>f : X \to Y</math> that is depicted below is neither total at <math>Y\!</math> nor tubular at <math>Y,\!</math> and so it cannot enjoy any of the properties of being surjective, injective, or bijective.

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

|}

|}

A cheap way of getting a surjective function out of any function is to reset its codomain to its range.  For example, the range of the function ''f'' above is ''Y''&prime;&nbsp;=&nbsp;{0,&nbsp;2,&nbsp;5,&nbsp;6,&nbsp;7,&nbsp;8,&nbsp;9}.  Thus, if we form a new function ''g'' : ''X'' &rarr; ''Y''&prime; that looks just like ''f'' on the domain ''X'' but is assigned the codomain ''Y''&prime;, then ''g'' is surjective, and is described as mapping "onto" ''Y''&prime;.

An easy way to extract a surjective function from any function is to reset its codomain to its range.  For example, the range of the function <math>f\!</math> above is <math>Y^\prime = \{ 0, 2, 5, 6, 7, 8, 9 \}.\!</math> Thus, if we form a new function <math>g : X \to Y^\prime</math> that looks just like <math>f\!</math> on the domain <math>X\!</math> but is assigned the codomain <math>Y^\prime,\!</math> then <math>g\!</math> is surjective, and is described as mapping ''onto'' <math>Y^\prime.\!</math>

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

|}

|}

The function ''h'' : ''Y''&prime; &rarr; ''Y'' is injective.

The function <math>h : Y^\prime \to Y</math> is injective.

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

|}

|}

The function ''m'' : ''X'' &rarr; ''Y'' is bijective.

The function <math>m : X \to Y</math> is bijective.

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"