Image of Intersection under Relation
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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:
- $\ds \RR \sqbrk {\bigcap \mathbb S} \subseteq \bigcap_{X \mathop \in \mathbb S} \RR \sqbrk X$
Family of Sets
Let $S$ and $T$ be sets.
Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.
Let $\RR \subseteq S \times T$ be a relation.
Then:
- $\ds \RR \sqbrk {\bigcap_{i \mathop \in I} S_i} \subseteq \bigcap_{i \mathop \in I} \RR \sqbrk {S_i}$
where $\ds \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\family {S_i}_{i \mathop \in I}$.
Proof
\(\ds S_1 \cap S_2\) | \(\subseteq\) | \(\ds S_1\) | Intersection is Subset | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \RR \sqbrk {S_1 \cap S_2}\) | \(\subseteq\) | \(\ds \RR \sqbrk {S_1}\) | Image of Subset is Subset of Image |
\(\ds S_1 \cap S_2\) | \(\subseteq\) | \(\ds S_2\) | Intersection is Subset | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \RR \sqbrk {S_1 \cap S_2}\) | \(\subseteq\) | \(\ds \RR \sqbrk {S_2}\) | Image of Subset is Subset of Image |
\(\ds \leadsto \ \ \) | \(\ds \RR \sqbrk {S_1 \cap S_2}\) | \(\subseteq\) | \(\ds \RR \sqbrk {S_1} \cap \RR \sqbrk {S_2}\) | Intersection is Largest Subset |
$\blacksquare$
Also see
- Image of Union under Relation
- Preimage of Union under Relation
- Preimage of Intersection under Relation
Also see
- Image of Intersection under One-to-Many Relation, which shows that, for the general relation $\RR$, equality holds if and only if $\RR$ is one-to-many.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations: Theorem $5 \ \text{(c)}$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.12$: Set Inclusions for Image and Inverse Image Sets: Theorem $12.2 \ \text{(b)}$