Set Union is Idempotent

Theorem

Set union is idempotent:

$S \cup S = S$

Proof

\(\ds x\)

\(\in\)

\(\ds S \cup S\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x \in S\)

\(\lor\)

\(\ds x \in S\)

Definition of Set Union

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds S\)

Rule of Idempotence: Disjunction

$\blacksquare$

Also see

• Set Intersection is Idempotent

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 8 \ \text{(a)}$
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Set Theory: $1.3$: Set operations
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B i}$
• 1968: A.A. Sveshnikov: Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions (translated by Richard A. Silverman) ... (previous) ... (next): $\text I$: Random Events: $1$. Relations among Random Events: Problem $1$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 6 \ \text{(c)}$
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.6$: Set Identities and Other Set Relations: Exercise $2 \ \text{(d)}$
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(a)}$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets: Exercise $1 \ \text{(ii)}$
• 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 $1.2.1 \ \text{(i)}$
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 2$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets: $\text {(v)}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): union: $\text {(i)}$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(v)}$