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Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:
Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms.
The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if and only if they have the same elements.