A
set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in
mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In
mathematics education, elementary topics such as
Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.
The elements of a set, also called its members, can be anything numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. The statement that sets A and B are equal means that they have precisely the same members (i.e., every member of A is also a member of B and vice versa).
Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple.
As discussed below, in formal mathematics the definition given above turned out to be inadequate; 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 they have the same elements.
