Axiom of Extensionality

Set Theory · Axiom Academy

LESSON Axiom of Extensionality Two sets are equal if and only if they have the same elements The Axiom of Extensionality is formally stated as follows: For all sets A and B, if every element of A is an element of B, and every element of B is an element of A, then A = B. In symbols, using the universal quantifier and logical implication: Let's see the axiom in action. Consider these two sets: Set A = 1, 2, 3 and Set B = 3, 2, 1 Are they equal? The Axiom of Extensionality tells us to check if they have exactly the same elements. Set C = 1, 2, 3 and Set D = 1, 2, 3, 4 These sets are NOT equal because D contains an element (4) that C does not have. The Axiom of Extensionality tells us that: Order doesn't matter: 1, 2, 3 = 3, 2, 1 Repetition doesn't matter: 1, 2, 2, 3 = 1, 2, 3 Only membership matters: Sets are completely determined by what's inside them

This is the written version of the interactive lesson above. See the full Set Theory course.