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Axiom: Axiom of Extensionality

If each element of the set \(X\) is also an element of the set \(Y\) and vice versa, then both are the same. In other words, a set is determined by its elements1, which is known as the extensionality principle.

\[\forall X~\forall Y (\forall z~(z\in X \Leftrightarrow z\in Y)\Rightarrow X=Y)\]

1 Please note that repeating the same elements in a set determines the same set.

| | | | | created: 2014-03-22 15:55:22 | modified: 2019-08-03 19:18:26 | by: bookofproofs | references: [656], [1038]

1.Explanation: What does the Extensionality Principle mean?

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[656] Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

[1038] Wille, D; Holz., M : “Repetitorium der Linearen Algebra”, Binomi Verlag, 1994