Intersection is Subset
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Theorem
The intersection of two sets is a subset of each:
- $S \cap T \subseteq S$
- $S \cap T \subseteq T$
General Result
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\mathbb S \subseteq \powerset S$.
Then:
- $\ds \forall T \in \mathbb S: \bigcap \mathbb S \subseteq T$
Family of Sets
In the context of a family of sets, the result can be presented as follows:
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Then:
- $\ds \forall \beta \in I: \bigcap_{\alpha \mathop \in I} S_\alpha \subseteq S_\beta$
where $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is the intersection of $\family {S_\alpha}_{\alpha \mathop \in I}$.
Proof
\(\ds x\) | \(\in\) | \(\ds S \cap T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds S \land x \in T\) | Definition of Set Intersection | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds S\) | Rule of Simplification | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds S \cap T\) | \(\subseteq\) | \(\ds S\) | Definition of Subset |
Similarly for $T$.
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 8 \ \text{(f)}$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.3$. Intersection: Example $13$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 5$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7$: Unions and Intersections
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets: Exercise $1.2.1 \ \text{(ii)}$