Axiom of Extensionality

The axiom of Zermelo-Fraenkel set theory which asserts that sets formed by the same elements are equal,

  forall x(x in a=x in b)=>a=b.

Note that some texts (e.g., Devlin 1993), use a bidirectional equivalent = preceding "a=b," while others (e.g., Enderton 1977, Itô 1986), use the one-way implies =>. However, one-way implication suffices.

Using the notation a subset b (a is a subset of b) for (x in a)=>(x in b), the axiom can be written concisely as

 a subset b ^ b subset a=>a=b,

where  ^ denotes logical AND.

See also

Zermelo-Fraenkel Set Theory

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Devlin, K. The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed. New York: Springer-Verlag, 1993.Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.Itô, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 147-148, 1986.

Referenced on Wolfram|Alpha

Axiom of Extensionality

Cite this as:

Weisstein, Eric W. "Axiom of Extensionality." From MathWorld--A Wolfram Web Resource.

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