Axiom:Axiom of Extension

Axiom

Two sets are equal iff they contain the same elements:

$\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

The order of the elements in the sets is immaterial.

Otherwise known as the Axiom of Extensionality or Axiom of Extent.

Notes

This is the fundamental definition of what a set is: a set is determined by its elements.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 1$: The Axiom of Extension
• W.E. Deskins: Abstract Algebra (1964): $\S 1.1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$

• Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html