Set (math)

In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed 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 German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.

A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the elements or members of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. 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:

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.

Cantor's definition turned out to be inadequate; instead, the notion of a "set" is taken as a primitive notion in axiomatic set theory, and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.

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