Composite with Constant Mapping is Constant Mapping
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Theorem
Let $f_c: S \to T$ be the constant mapping defined as:
- $\forall x \in S: \map {f_c} x = c$
where $c \in T$.
Then for all mappings $g: \Dom g \to S$:
- $f_c \circ g$ is a constant mapping
and for all mappings $h: T \to \Cdm h$:
- $h \circ f_c$ is a constant mapping
where:
- $\Dom g$ denotes the domain of $g$
- $\Cdm h$ denotes the codomain of $h$
- $\circ$ denotes composition of mappings.
Proof
\(\ds \forall x \in \Dom g: \, \) | \(\ds \map {\paren {f_c \circ g} } x\) | \(=\) | \(\ds \map {f_c} {\map g x}\) | Definition of Composition of Mappings | ||||||||||
\(\ds \) | \(=\) | \(\ds c\) | Definition of Constant Mapping |
$\Box$
\(\ds \forall x \in S: \, \) | \(\ds \map {\paren {h \circ f_c} } x\) | \(=\) | \(\ds \map h {\map {f_c} x}\) | Definition of Composition of Mappings | ||||||||||
\(\ds \) | \(=\) | \(\ds \map h c\) | Definition of Constant Mapping |
As $c$ is constant, so is $\map h c$.
$\blacksquare$