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One can take the union of several sets simultaneously. 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 ...
For instance, for the sets {1, 2, 3} and {2, 3, 4}, the symmetric difference set is {1, 4}. It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A). Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a, b), where a is a member of A and b is a ...
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
(The notation (a, b) is also used to denote an open interval on the real number line, but the context should make it clear which meaning is intended. Otherwise, the notation ]a, b[ may be used to denote the open interval whereas (a, b) is used for the ordered pair). If A and B are sets, then the Cartesian product (or simply product) is defined ...
The complement of the union of two sets is the same as the intersection of their complements; 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 ...
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".
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 ...
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.