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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.
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
Alternative set theories include: [1] Vopěnka's alternative set theory; Von Neumann–Bernays–Gödel set theory; Morse–Kelley set theory; Tarski–Grothendieck set theory
If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
When A is a subset of U, the set difference U \ A is also called the complement of A in U. 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.
The set B doesn't have to be a subset of A for A \ B to make sense; this is the difference between the relative complement and the absolute complement (A C = U \ A) from the previous section. To illustrate these ideas, let A be the set of left-handed people, and let B be the set of people with blond hair.
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 concept of a universe is reflected in the use of Venn diagrams. In a Venn diagram, the action traditionally takes place inside a large rectangle that represents the universe U. One generally says that sets are represented by circles; but these sets can only be subsets of U. The complement of a set A is then given by that portion of the ...