# Bijection/Examples

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## Examples of Bijections

### Arbitrary Mapping on Sets

Let $A = \set {a_1, a_2, a_3, a_4}$.

Let $B = \set {b_1, b_2, b_3, b_4}$.

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

- $f = \set {\tuple {a_1, b_3}, \tuple {a_2, b_2}, \tuple {a_3, b_4}, \tuple {a_4, b_1} }$

Then $f$ is a bijection.

### $\paren {-1}^x \floor {\dfrac x 2}$ from $\N$ to $\Z$

Let $f: \N \to \Z$ be the mapping defined from the natural numbers to the integers as:

- $\forall x \in \N: f \paren x = \paren {-1}^x \floor {\dfrac x 2}$

Then $f$ is a bijection.

### $x^3$ Function on Real Numbers is Bijective

Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:

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

Then $f$ is a bijection.

### Negative Functions on Standard Number Systems are Bijective

Let $\mathbb S$ be one of the standard number systems $\Z$, $\Q$, $\R$, $\C$. Let $h: \mathbb S \to \mathbb S$ be the negation function defined on $\mathbb S$:

- $\forall x \in \mathbb S: \map h x = -x$

Then $h$ is a bijection.

### $2 x + 1$ Function on Real Numbers is Bijective

Let $f: \R \to \R$ be the mapping defined on the set of real numbers as:

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

Then $f$ is a bijection.

### $n + 1$ Mapping on Integers is Bijective

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

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

Then $f$ is a bijection.