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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 ...
It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.
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.
When is empty, the condition given above is an example of a vacuous truth. 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.
The union of sets is distributive over intersection, and intersection is distributive over union. ... In standard truth-functional propositional logic, ...
De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
A dot in the diagram above a slanting line indicates truth for that statement; likewise, a dot below indicates falsity. The R-diagrams for p and q are shown below, respectively: For more than two statements, the four spaces formed by the intersection of lines p and q must be subdivided into more lines.
A term's definition may require additional properties that are not listed in this table. This Hasse diagram depicts a partially ordered set with four elements: a , b , the maximal element a ∨ {\displaystyle \vee } b equal to the join of a and b , and the minimal element a ∧ {\displaystyle \wedge } b equal to the meet of a and b .