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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 ...
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory , and to illustrate simple set relationships in probability , logic , statistics , linguistics and computer science .
Venn diagram of = . The symmetric difference is equivalent to the union of both relative complements, that is: [1] = (), The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation:
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.
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
In this case, if the choice of U is clear from the context, the notation A c is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams. Symmetric difference of sets A and B, denoted A B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of ...
In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a ...
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".