Union with Superset is Superset
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Theorem
- $S \subseteq T \iff S \cup T = T$
where:
Proof 1
Let $S \cup T = T$.
Then by definition of set equality:
- $S \cup T \subseteq T$
Thus:
\(\ds S\) | \(\subseteq\) | \(\ds S \cup T\) | Subset of Union | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds T\) | Subset Relation is Transitive |
Now let $S \subseteq T$.
From Subset of Union, we have $S \cup T \supseteq T$.
We also have:
\(\ds S\) | \(\subseteq\) | \(\ds T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds S \cup T\) | \(\subseteq\) | \(\ds T \cup T\) | Set Union Preserves Subsets | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds S \cup T\) | \(\subseteq\) | \(\ds T\) | Set Union is Idempotent |
Then:
\(\ds S \cup T\) | \(\subseteq\) | \(\ds T\) | ||||||||||||
\(\ds S \cup T\) | \(\supseteq\) | \(\ds T\) |
By definition of set equality:
- $S \cup T = T$
So:
\(\ds S \cup T = T\) | \(\implies\) | \(\ds S \subseteq T\) | ||||||||||||
\(\ds S \subseteq T\) | \(\implies\) | \(\ds S \cup T = T\) |
and so:
- $S \subseteq T \iff S \cup T = T$
from the definition of equivalence.
$\blacksquare$
Proof 2
\(\ds \) | \(\) | \(\ds S \cup T = T\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \paren {x \in S \lor x \in T \iff x \in T}\) | Definition of Set Equality | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds \paren {x \in S \implies x \in T}\) | Conditional iff Biconditional of Consequent with Disjunction | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \) | \(\) | \(\ds S \subseteq T\) | Definition of Subset |
$\blacksquare$
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 $\text{B} \ 1 \ \text{(a), (c)}$
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Exercise $1 \ \text {(c)}$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.3 \ \text{(d)}$
- 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: $\S 6 \ \text{(d)}$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \ \text{(a)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7.3 \ \text{(i)}$: 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 $1.2.1 \ \text{(vii)}$
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- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.11$