Image of Intersection under One-to-Many Relation/General Result

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Theorem

Let $S$ and $T$ be sets.

Let $\RR \subseteq S \times T$ be a relation.

Let $\powerset S$ be the power set of $S$.


Then:

$\ds \forall \mathbb S \subseteq \powerset S: \RR \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

if and only if $\RR$ is one-to-many.


Proof

Sufficient Condition

Suppose:

$\ds \RR \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

where $\mathbb S$ is any subset of $\powerset S$.

Then by definition of $\mathbb S$:

$\forall S_1, S_2 \in \mathbb S: \RR \sqbrk {S_1 \cap S_2} = \RR \sqbrk {S_1} \cap \RR \sqbrk {S_2}$

and the sufficient condition applies for Image of Intersection under One-to-Many Relation.

So $\RR$ is one-to-many.

$\Box$


Necessary Condition

Suppose $\RR$ is one-to-many.


From Image of Intersection under Relation: General Result, we already have:

$\ds \RR \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

so we just need to show:

$\ds \forall \mathbb S \subseteq \powerset S: \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X \subseteq \RR \sqbrk {\bigcap \mathbb S}$


Let:

$\ds t \in \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

Then:

\(\ds t\) \(\in\) \(\ds \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X\)
\(\ds \leadsto \ \ \) \(\ds \forall X \in \mathbb S: \, \) \(\ds t\) \(\in\) \(\ds \RR \sqbrk X\) Definition of Set Intersection
\(\ds \leadsto \ \ \) \(\ds \forall X \in \mathbb S: \exists x \in X: \, \) \(\ds \tuple {x, t}\) \(\in\) \(\ds \RR\) Definition of Relation
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \bigcap \mathbb S\) Definition of One-to-Many Relation
\(\ds \leadsto \ \ \) \(\ds \RR \sqbrk x\) \(\in\) \(\ds \RR \sqbrk {\bigcap \mathbb S}\) Image of Element is Subset
\(\ds \leadsto \ \ \) \(\ds \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X\) \(\subseteq\) \(\ds \RR \sqbrk {\bigcap \mathbb S}\) Definition of Subset


So if $\RR$ is one-to-many, it follows that:

$\ds \forall \mathbb S \subseteq \powerset S: \RR \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

$\Box$


Putting the results together:

$\RR$ is one-to-many if and only if:

$\ds \RR \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$

where $\mathbb S$ is any subset of $\powerset S$.

$\blacksquare$