Definition:Bijective Restriction

Definition

Let $f: S \to T$ be a mapping which is not bijective.

A bijective restriction of $f$ is a restriction $f {\restriction_{S' \times T'} }: S' \to T'$ of $f$ such that $f {\restriction_{S' \times T'} }$ is a bijection.

Examples

Bijective Restrictions of $f \paren x = x^2 - 4 x + 5$

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

$\forall x \in \R: \map f x = x^2 - 4 x + 5$

The following real functions are bijective restrictions of $f$:

\(\ds f_1: \hointl \gets 2\)

\(\to\)

\(\ds \hointr 1 \to\)

\(\ds f_2: \hointr 2 \to\)

\(\to\)

\(\ds \hointr 1 \to\)

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 23$: Restriction of a Mapping