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  2. List of set identities and relations - Wikipedia

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

    In the left hand sides of the following identities, L is the L eft most set and R is the R ight most set. Whenever necessary, both L and R should be assumed to be subsets of some universe set X , so that L ∁ := X ∖ L and R ∁ := X ∖ R . {\displaystyle L^{\complement }:=X\setminus L{\text{ and }}R^{\complement }:=X\setminus R.}

  3. Algebra of sets - Wikipedia

    en.wikipedia.org/wiki/Algebra_of_sets

    The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".

  4. Relation (mathematics) - Wikipedia

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

    In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...

  5. Intersection (set theory) - Wikipedia

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

    So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...

  6. List of types of sets - Wikipedia

    en.wikipedia.org/wiki/List_of_types_of_sets

    List of set identities and relations – Equalities for combinations of sets; List of types of functions This page was last edited on 20 April 2024, at 21:36 ...

  7. Complement (set theory) - Wikipedia

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

    If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...

  8. Union (set theory) - Wikipedia

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

    In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. [1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero (⁠ ⁠) sets and it is by definition equal to the empty set.

  9. Transitive relation - Wikipedia

    en.wikipedia.org/wiki/Transitive_relation

    As are the set of real numbers or the set of natural numbers: whenever x > y and y > z, then also x > z whenever x ≥ y and y ≥ z, then also x ≥ z whenever x = y and y = z, then also x = z. More examples of transitive relations: "is a subset of" (set inclusion, a relation on sets) "divides" (divisibility, a relation on natural numbers)