Explanation: What does the Extensionality Principle mean?

(related to Axiom: Axiom of Extensionality)

Sets are +determined+ by the elements they contain. This is known as the extensionality principle and means that two sets \(A\) and \(B\) are equal (i.e. are indistinguishable), if and only if they contain the same elements.

In other words, the extensionality principle means that if you try to assign additional properties to sets, for instance, talk about their "colors" like an "azure" set \(A:=\left\{ x,y \right\}\) and a "black" set \(B:=\left\{x,y\right\}\), then this would not make any sense, since both sets contain the same elements \(x\) and \(y\) and so they are equal \(A=B\).

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  2. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994