Definition:Bijective Restriction
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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