Complement of Direct Image Mapping of Injection equals Direct Image of Complement

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Theorem

Let $f: S \to T$ be an injection.

Let $f^\to: \powerset S \to \powerset T$ denote the direct image mapping of $f$.


Then:

$\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} } A$

where $\circ$ denotes composition of mappings.


Proof

As $f$ is an injection, it is a fortiori a one-to-many relation.


From Image of Set Difference under Relation: Corollary 2:

$\forall A \in \powerset S: \map {\paren {\complement_{\Img \RR} \circ \RR^\to} } A = \map {\paren {\RR^\to \circ \complement_S} } A$

where $\RR \subseteq S \times T$ is a one-to-many relation on $S \times T$.


Substituting $f$ for $\RR$ gives the result:

$\forall A \in \powerset S: \map {\paren {\complement_{\Img f} \circ f^\to} } A = \map {\paren {f^\to \circ \complement_S} } A$

$\blacksquare$


Sources