Composite of Inverses of Induced Mappings

Theorem

Let $X, Y, Z$ be sets.

Let $f: X \to Y$ and $g: Y \to Z$ be mappings.

Let $f^\to: \mathcal P \left({X}\right) \to \mathcal P \left({Y}\right)$ and $g^\to: \mathcal P \left({Y}\right) \to \mathcal P \left({Z}\right)$ be the mappings induced by $f$ and $g$ on the power sets of $X$ and $Y$.

Let $f^\gets: \mathcal P \left({Y}\right) \to \mathcal P \left({X}\right)$ and $g^\gets: \mathcal P \left({Z}\right) \to \mathcal P \left({Y}\right)$ be the inverses of $f^\to$ and $g^\to$.

Then:

$f^\gets \circ g^\gets: P \left({Z}\right) \to P \left({X}\right)$ is the inverse of $g^\to \circ f^\to$

where $\circ$ denotes composition of mappings.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 10$: Inverses and Composites