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: , 
 Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011
 Wille, D; Holz., M : “Repetitorium der Linearen Algebra”, Binomi Verlag, 1994