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.


$2 x + 1$ 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.