Direct Image of Inverse Image of Direct Image equals Direct Image Mapping
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Theorem
Let $f: S \to T$ be a mapping.
Let:
- $f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$
- $f^\gets: \powerset T \to \powerset S$ denote the inverse image mapping of $f$
where $\powerset S$ denotes the power set of $S$.
Then:
- $f^\to \circ f^\gets \circ f^\to = f^\to$
where $\circ$ denotes composition of mappings.
Proof
\(\ds \forall A \in \powerset S: \, \) | \(\ds A\) | \(\subseteq\) | \(\ds \map {\paren {f^\gets \circ f^\to} } A\) | Subset of Domain is Subset of Preimage of Image | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall A \in \powerset S: \, \) | \(\ds \map {f^\to} A\) | \(\subseteq\) | \(\ds \map {\paren {f^\to \circ f^\gets \circ f^\to} } A\) | Image of Subset under Mapping is Subset of Image | ||||||||||
and: | |||||||||||||||
\(\ds \forall B \in \powerset T: \, \) | \(\ds \map {\paren {f^\to \circ f^\gets} } B\) | \(\subseteq\) | \(\ds B\) | Subset of Codomain is Superset of Image of Preimage | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall A \in \powerset S: \, \) | \(\ds \map {\paren {f^\to \circ f^\gets} } {\map {f^\to} A}\) | \(\subseteq\) | \(\ds \map {f^\to} A\) | Image of Subset under Mapping is Subset of Image, setting $B = \map {f^\to} A \in \powerset T$ |
Thus we have:
- $\forall A \in \powerset S: \map {f^\to} A \subseteq \map {\paren {f^\to \circ f^\gets \circ f^\to} } A \subseteq \map {f^\to} A$
and the result follows.
$\blacksquare$
Also see
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $4$