Union is Idempotent

Theorem

Set union is idempotent:

$S \cup S = S$

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle S \cup S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\lor\)	\(\displaystyle x \in S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of set union
	\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Idempotence

$\blacksquare$

Also see

• Intersection is Idempotent

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 4$: Unions and Intersections
• W.E. Deskins: Abstract Algebra (1964): $\S 1.1$: Exercise $1.1: \ 8 \ \text{(a)}$
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$: Exercises $\text{B i}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 6 \ \text{(c)}$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.2$: Exercise $1.2.1 \ \text{(i)}$