axiom of extensionality

If $X$ and $Y$ have the same elements, then $X=Y$.

The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads:

 $\forall u(u\in X\leftrightarrow u\in Y)\rightarrow X=Y.$

Note that the converse,

 $X=Y\rightarrow\forall u(u\in X\leftrightarrow u\in Y)$

is an axiom of the predicate calculus. Hence we have,

 $X=Y\leftrightarrow\forall u(u\in X\leftrightarrow u\in Y).$

Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.

Title axiom of extensionality AxiomOfExtensionality 2013-03-22 13:42:40 2013-03-22 13:42:40 Sabean (2546) Sabean (2546) 5 Sabean (2546) Axiom msc 03E30 extensionality