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Usually, we denote sets by capital letters $X,Y,\ldots$, while writing small letters $x,y,\ldots$ for their elements. In the Zermelo-Fraenkel set theory, sets can be themselves elements of other sets. Thus, there is no reason to introduce a different notation for sets and their elements and many sources in literature do without a distinctive notation. Nevertheless, we will keep sticking to this notation, just to make it clearer which sets are meant to be the elements and which sets are meant to contain these elements. When you read small letters $x,y,\ldots$ below, please keep in mind that they denote sets again.
Axiom: Zermelo-Fraenkel Axioms
A set is a mathematical object fulfilling the following axioms: