Restriction/Mapping/Examples

Examples of Restrictions of Mappings

Restriction of Square Function on Natural Numbers

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

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

Let $S = \set {x \in \N: \exists y \in \N_{>0}: x = 2 y} = \set {2, 4, 6, 8, \ldots}$

Let $g: S \to \N$ be the mapping defined as:

$\forall n \in \N: \map g n = n^2$

Then $g$ is a restriction of $f$.

Bijective Restriction of Real Sine Function

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

$\forall x \in \R: f \paren x = \sin x$

Then a bijective restriction $g$ of $f$ can be defined as:

$g: S \to T: \forall x \in S: g \paren x = \sin x$

where:

$S = \closedint {-\dfrac \pi 2} {\dfrac \pi 2}$

$T = \closedint {-1} 1$