Image of Intersection under Relation/Family of Sets
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Theorem
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
\(\ds \forall i \in I: \, \) | \(\ds \bigcap_{j \mathop \in I} S_j\) | \(\subseteq\) | \(\ds S_i\) | Intersection is Subset: Family of Sets | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i \in I: \, \) | \(\ds \RR \sqbrk {\bigcap_{j \mathop \in I} S_j}\) | \(\subseteq\) | \(\ds \RR \sqbrk {S_i}\) | Image of Subset is Subset of Image | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \RR \sqbrk {\bigcap_{i \mathop \in I} S_i}\) | \(\subseteq\) | \(\ds \bigcap_{i \mathop \in I} \RR \sqbrk {S_i}\) | Intersection is Largest Subset: Family of Sets |
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations: Theorem $5 \ \text{(d)}$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.12$: Set Inclusions for Image and Inverse Image Sets: Theorem $12.5 \ \text{(b)}$