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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 .
Fundamentals. 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".
Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, a branch of mathematics, 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
Existential generalization / instantiation. 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. They are named after Augustus De Morgan, a 19th-century British mathematician.
Euler diagram. An Euler diagram (/ ˈɔɪlər /, OY-lər) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Venn diagrams.
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 ...
Statement. The symmetric difference is the set of elements that are in either set, but not in the intersection. Symbolic statement. A Δ B = ( A ∖ B ) ∪ ( B ∖ A ) {\displaystyle A\,\Delta \,B=\left (A\setminus B\right)\cup \left (B\setminus A\right)} In mathematics, the symmetric difference of two sets, also known as the disjunctive union ...
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. [ 1 ] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero ( ) sets and it is by definition equal to the empty set.