# Image of Intersection under Relation

## Theorem

Let $S$ and $T$ be sets.

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

Let $S_1$ and $S_2$ be subsets of $S$.

Then:

$\RR \sqbrk {S_1 \cap S_2} \subseteq \RR \sqbrk {S_1} \cap \RR \sqbrk {S_2}$

That is, the image of the intersection of subsets of $S$ is a subset of the intersection of their images.

### General Result

Let $S$ and $T$ be sets.

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

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

Let $\mathbb S \subseteq \powerset S$.

Then:

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

### Family of Sets

Let $S$ and $T$ be sets.

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.

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

Then:

$\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} S_i}\right] \subseteq \bigcap_{i \mathop \in I} \mathcal R \left[{S_i}\right]$

where $\displaystyle \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\left\langle{S_i}\right\rangle_{i \in I}$.

## Proof

 $\displaystyle S_1 \cap S_2$ $\subseteq$ $\displaystyle S_1$ Intersection is Subset $\displaystyle \leadsto \ \$ $\displaystyle \RR \sqbrk {S_1 \cap S_2}$ $\subseteq$ $\displaystyle \RR \sqbrk {S_1}$ Image of Subset is Subset of Image

 $\displaystyle S_1 \cap S_2$ $\subseteq$ $\displaystyle S_2$ Intersection is Subset $\displaystyle \leadsto \ \$ $\displaystyle \RR \sqbrk {S_1 \cap S_2}$ $\subseteq$ $\displaystyle \RR \sqbrk {S_2}$ Image of Subset is Subset of Image

 $\displaystyle \leadsto \ \$ $\displaystyle \RR \sqbrk {S_1 \cap S_2}$ $\subseteq$ $\displaystyle \RR \sqbrk {S_1} \cap \RR \sqbrk {S_2}$ Intersection is Largest Subset

$\blacksquare$