**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 elements^{1}, 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: 2018-03-22 23:27:47 | by: *bookofproofs* | references: [656], [1038]

## 1.**Explanation**: What does the Extensionality Principle mean?

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[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

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