Definition:Set Equality
From ProofWiki
Definition
Two sets are equal if and only if they have the same elements.
This can be defined rigorously as:
- $S = T \iff \left({\forall x: x \in S \iff x \in T}\right)$
where $S$ and $T$ are both sets.
Axiomatic Set Theory
The concept of set equality is axiomatised in the Axiom of Extension in Zermelo-Fraenkel set theory.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 1$: The Axiom of Extension
- Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 1$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 2$
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 2 \ \text{(f)}$
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
