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  2. Set (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Set_(mathematics)

    A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...

  3. Set-theoretic definition of natural numbers - Wikipedia

    en.wikipedia.org/wiki/Set-theoretic_definition...

    The set N of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The structure N, 0, S is a model of the Peano axioms (Goldrei 1996). The existence of the set N is equivalent to the axiom of infinity in ZF set theory.

  4. List of axioms - Wikipedia

    en.wikipedia.org/wiki/List_of_axioms

    Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...

  5. Naive set theory - Wikipedia

    en.wikipedia.org/wiki/Naive_set_theory

    It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. [4] Sets are of great importance in mathematics; in modern formal treatments, most mathematical ...

  6. Class (set theory) - Wikipedia

    en.wikipedia.org/wiki/Class_(set_theory)

    Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers.

  7. Paradoxes of set theory - Wikipedia

    en.wikipedia.org/wiki/Paradoxes_of_set_theory

    Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions.

  8. Set theory - Wikipedia

    en.wikipedia.org/wiki/Set_theory

    Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.

  9. List of set identities and relations - Wikipedia

    en.wikipedia.org/wiki/List_of_set_identities_and...

    This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.