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

## Theorem

Let $S$ and $T$ be sets.

Let $\mathcal R \subseteq S \times T$ be a relation.

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

Then:

$\displaystyle \forall \mathbb S \subseteq \powerset S: \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

if and only if $\mathcal R$ is one-to-many.

## Proof

### Sufficient Condition

Suppose:

$\displaystyle \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

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

Then by definition of $\mathbb S$:

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

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

So $\mathcal R$ is one-to-many.

$\Box$

### Necessary Condition

Suppose $\mathcal R$ is one-to-many.

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

$\displaystyle \mathcal R \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

so we just need to show:

$\displaystyle \forall \mathbb S \subseteq \powerset S: \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X \subseteq \mathcal R \sqbrk {\bigcap \mathbb S}$

Let:

$\displaystyle t \in \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

Then:

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

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

$\displaystyle \forall \mathbb S \subseteq \powerset S: \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

$\Box$

Putting the results together:

$\mathcal R$ is one-to-many if and only if:

$\displaystyle \mathcal R \sqbrk {\bigcap \mathbb S} = \bigcap_{X \mathop \in \mathbb S} \mathcal R \sqbrk X$

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

$\blacksquare$