Image of Intersection under Injection/General Result
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Theorem
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let $\powerset S$ be the power set of $S$.
Then:
- $\ds \forall \mathbb S \subseteq \powerset S: f \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} f \sqbrk X$
if and only if $f$ is an injection.
Proof
An injection is a type of one-to-one relation, and therefore also a one-to-many relation.
Therefore Image of Intersection under One-to-Many Relation applies:
- $\ds \forall \mathbb S \subseteq \powerset S: \RR \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk {\mathbb S}$
if and only if $\RR$ is a one-to-many relation.
We have that $f$ is a mapping and therefore a many-to-one relation.
So $f$ is a one-to-many relation if and only if $f$ is also an injection.
It follows that:
- $\ds \forall \mathbb S \subseteq \powerset S: f \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} f \sqbrk X$
if and only if $f$ is an injection.
$\blacksquare$
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions: Exercise $7 \ \text {(b)}$