Formal Statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
- Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.
(It is not really essential that C here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)
The converse, of this axiom follows from the substitution property of equality.
Read more about this topic: Axiom Of Extensionality
Famous quotes containing the words formal and/or statement:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“Most personal correspondence of today consists of letters the first half of which are given over to an indexed statement of why the writer hasn’t written before, followed by one paragraph of small talk, with the remainder devoted to reasons why it is imperative that the letter be brought to a close.”
—Robert Benchley (1889–1945)