Bijection/Examples

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.

$a x + b$ Function on Real Numbers is Bijective

Bijection/Examples/ax+b Function on Real Numbers

$n + 1$ 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.