# Injection/Examples

## Examples of Injections

### $2 x$ Function on Integers is Injective but not Surjective

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

- $\forall x \in \Z: \map f x = 2 x$

Then $f$ is an injection, but not a surjection.

### $2 x + 1$ Function on Integers is Injective

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

- $\forall x \in \Z: \map f x = 2 x + 1$

Then $f$ is an injection.

### $-x$ Function on Integers is Injective

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

- $\forall x \in \Z: \map f x = -x$

Then $f$ is an injection.

### Square Function on $\N$ is Injective

Let $f: \N \to \N$ be the mapping defined as:

- $\forall n \in \N: \map f n = n^2$

Then $f$ is an injection, but not a surjection.

### Cube Function is Injective

Let $f: \R \to \R$ be the real function defined as:

- $\forall x \in \R: \map f x = x^3$

Then $f$ is an injection.

## Examples of Mappings which are Not Injections

### Arbitrary Mapping on Sets

Let $A = \set {a, b, c, d}$.

Let $B = \set {1, 2, 3, 4, 5}$.

Let $f \subseteq {A \times B}$ be the mapping defined as:

- $f = \set {\tuple {a, 1}, \tuple {b, 4}, \tuple {c, 5}, \tuple {d, 4} }$

Then $f$ is not a injection.

### Square Function is Not Injective

Let $f: \R \to \R$ be the real square function:

- $\forall x \in \R: \map f x = x^2$

Then $f$ is not an injection.

### $\map f x = \dfrac x 2$ for $x$ Even, $0$ for $x$ Odd is Not Injective

Let $f: \Z \to Z$ be the real function defined as:

- $\forall x \in \Z: \map f x = \begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$

Then $f$ is not an injection.