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Set (mathematics)


This article is about what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see Naive set theory. For a rigorous modern axiomatic treatment of sets, see Set theory.
A set of polygons in an Euler diagram

A set is the mathematical model for a collection of different[1] things;[2][3][4] a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.[5] The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.[6]

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.[5]

  1. ^ Cite error: The named reference Cantor was invoked but never defined (see the help page).
  2. ^ P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995). Functional Analysis. New Age International. p. 1. ISBN 978-81-224-0801-0.
  3. ^ Samuel Goldberg (1 January 1986). Probability: An Introduction. Courier Corporation. p. 2. ISBN 978-0-486-65252-8.
  4. ^ Thomas H. Cormen; Charles E Leiserson; Ronald L Rivest; Clifford Stein (2001). Introduction To Algorithms. MIT Press. p. 1070. ISBN 978-0-262-03293-3.
  5. ^ a b Halmos 1960, p. 1.
  6. ^ Stoll, Robert (1974). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5. ISBN 9780716704577.

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