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If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts and spades. When the universe is the universe of sets described in formalized set theory , the absolute complement of a set is generally not itself a set, but rather a proper class .
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. The properties below are stated without proof , but can be derived from a small number of properties taken as axioms .
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".
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 ...
u(a, b) = u(b, a) Axiom u4. Associativity u(a, u(b, d)) = u(u(a, b), d) Axiom u5. Continuity u is a continuous function Axiom u6. Superidempotency u(a, a) > a for all 0 < a < 1 Axiom u7. Strict monotonicity a 1 < a 2 and b 1 < b 2 implies u(a 1, b 1) < u(a 2, b 2) Axioms u1 up to u4 define a t-conorm (aka s-norm or fuzzy union). The standard t ...
the set difference A \ B (also written A − B) is the set of all things that belong to A but not B. Especially when B is a subset of A, it is also called the relative complement of B in A. With B c as the absolute complement of B (in the universal set U), A \ B = A ∩ B c. their symmetric difference A Δ B is the set of all things that belong ...
That is, for any sets ,, and , one has = () = () Inside a universe , one may define the complement of to be the set of all elements of not in . Furthermore, the intersection of A {\displaystyle A} and B {\displaystyle B} may be written as the complement of the union of their complements, derived easily from De Morgan's laws : A ∩ B = ( A c ...
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...