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  2. 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 ...

  3. De Morgan's laws - Wikipedia

    en.wikipedia.org/wiki/De_Morgan's_laws

    The complement of the intersection of two sets is the same as the union of their complements; or not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one of A or B. De Morgan's law with set subtraction operation

  4. List of set identities and relations - Wikipedia

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

    Universe set and complement notation The notation L ∁ = def X ∖ L . {\displaystyle L^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus L.} may be used if L {\displaystyle L} is a subset of some set X {\displaystyle X} that is understood (say from context, or because it is clearly stated what the superset X ...

  5. 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".

  6. Union (set theory) - Wikipedia

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

    In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula = (), where the superscript denotes the complement in the universal set ⁠ ⁠. Alternatively, intersection can be expressed in terms of union and complementation in a similar way: A ∩ B = ( A ∁ ∪ B ∁ ) ∁ {\displaystyle A\cap B ...

  7. Simple theorems in the algebra of sets - Wikipedia

    en.wikipedia.org/wiki/Simple_theorems_in_the...

    P(U) is assumed closed under union, intersection, and set complement. The algebra of sets is an interpretation or model of Boolean algebra, with union, intersection, set complement, U, and {} interpreting Boolean sum, product, complement, 1, and 0, respectively.

  8. 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 ...

  9. Naive set theory - Wikipedia

    en.wikipedia.org/wiki/Naive_set_theory

    It is denoted by A ∪ B. The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B. Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A \ B or A − B.