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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 .
In set theory, a universal set is a set which contains all objects, including itself. [1] In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set.
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 ...
English: Venn diagram for the set theoretic intersection of A and B. Français : Diagramme de Venn montrant l'intersection de deux ensembles A et B. Italiano: Diagramma di Venn per l'intersezione degli insiemi A e B.
For instance, when investigating properties of the real numbers R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories. Given a universal set U and a subset A of U, the complement of A (in U) is defined as
The only sets that Cantor was originally interested in were subsets of R. 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 ...
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 ...
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 ...
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