Intersection with Subset is Subset
Jump to navigation
Jump to search
Theorem
- $S \subseteq T \iff S \cap T = S$
where:
- $S \subseteq T$ denotes that $S$ is a subset of $T$
- $S \cap T$ denotes the intersection of $S$ and $T$.
Proof
Let $S \cap T = S$.
Then by the definition of set equality, $S \subseteq S \cap T$.
Thus:
\(\ds S \cap T\) | \(\subseteq\) | \(\ds T\) | Intersection is Subset | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds T\) | Subset Relation is Transitive |
Now let $S \subseteq T$.
From Intersection is Subset we have $S \supseteq S \cap T$.
We also have:
\(\ds S\) | \(\subseteq\) | \(\ds T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds S \cap S\) | \(\subseteq\) | \(\ds T \cap S\) | Set Intersection Preserves Subsets | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds S \cap T\) | Intersection is Idempotent and Intersection is Commutative |
So as we have:
\(\ds S \cap T\) | \(\subseteq\) | \(\ds S\) | ||||||||||||
\(\ds S \cap T\) | \(\supseteq\) | \(\ds S\) |
it follows from the definition of set equality that:
- $S \cap T = S$
So we have:
\(\ds S \cap T = S\) | \(\implies\) | \(\ds S \subseteq T\) | ||||||||||||
\(\ds S \subseteq T\) | \(\implies\) | \(\ds S \cap T = S\) |
and so:
- $S \subseteq T \iff S \cap T = S$
from the definition of equivalence.
$\blacksquare$
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $1$
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 9$
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{B} \ 1 \ \text{(a), (b)}$
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Exercise $1 \ \text {(d)}$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $2$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $3.3 \ \text{(c)}$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B iv}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 5 \ \text{(d)}$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.6$: Set Identities and Other Set Relations: Exercise $1 \ \text{(c)}$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(f)}$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \ \text{(b)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7.3 \ \text {(ii)}$: 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 $1.2.1 \ \text{(vii)}$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Exercise $5.6. \ \text {(e)}$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.11$