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Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects).
In Political science and Decision theory, order relations are typically used in the context of an agent's choice, for example the preferences of a voter over several political candidates. x ≺ y means that the voter prefers candidate y over candidate x. x ~ y means the voter is indifferent between candidates x and y.
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.
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 ...
Given a set X, a relation R over X is a set of ordered pairs of elements from X, formally: R ⊆ { (x,y) | x, y ∈ X}. [2] [10] The statement (x,y) ∈ R reads "x is R-related to y" and is written in infix notation as xRy. [7] [8] The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy.
The set of all ordered pairs obtained from two sets, where each pair consists of one element from each set. cardinal 1. A cardinal number is an ordinal with more elements than any smaller ordinal cardinality The number of elements of a set categorical 1. A theory is called categorical if all models are isomorphic. This definition is no longer ...
With a dot over it, it is the class of ordered pairs. With the subscript r it is the class of unequal ordered pairs. *54.02, *56.01, *56.02 An ordered pair *55.01 Cl Short for "class". The powerset relation *60.01 Cl ex The relation saying that one class is the set of non-empty classes of another *60.02 Cls 2, Cls 3
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.