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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.
In psychology, a set is a group of expectations that shape experience by making people especially sensitive to specific kinds of information. A perceptual set, also called perceptual expectancy, is a predisposition to perceive things in a certain way. [1] Perceptual sets occur in all the different senses. [2]
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.
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.
The above concept of relation [a] has been generalized to admit relations between members of two different sets (heterogeneous relation, like "lies on" between the set of all points and that of all lines in geometry), relations between three or more sets (finitary relation, like "person x lives in town y at time z "), and relations between ...
In mathematics and logic, relations are defined as set-theoretic structures. For example, the relation less than is defined as the set of all ordered pairs in which the first element is less than the second element. This set includes pairs like [1,2], [1,3], and [2,17]. [9]
The relation "is the birth parent of" on a set of people is not a transitive relation. However, in biology the need often arises to consider birth parenthood over an arbitrary number of generations: the relation "is a birth ancestor of" is a transitive relation and it is the transitive closure of the relation "is the birth parent of".
In set theory, a standard model for a theory T is a model M for T where the membership relation ∈ M is the same as the membership relation ∈ of a set theoretical universe V (restricted to the domain of M). In other words, M is a substructure of V.