an ordered pair, denoted (a, b), is a pair of objects in which their order is significant. The ordered pair (a, b) is different from the ordered pair (b...

reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )} consisting...

also defined from ordered pairs by a recurrence starting from an ordered pair; indeed, an n-tuple can be identified with the ordered pair of its (n − 1)...

in P {\displaystyle P} or P − {\displaystyle P^{-}} , for example the ordered pair (0,0). Then the integers are defined to be the union P ∪ P − ∪ { 0 }...

{\displaystyle x\cup x^{c}=V} . Ordered Pair: For each a {\displaystyle a} , b {\displaystyle b} , the ordered pair of a {\displaystyle a} and b {\displaystyle...

in the chain. Thus a singleton set is a chain of length zero, and an ordered pair is a chain of length one. The dimension of a space is often defined or...

by pairs of real numbers x and y. The order of the components x and y is significant, so such a pair is also called an ordered pair. Such a pair is written...

that, two ordered pairs are equal if and only if their first elements are equal and their second elements are equal. Formally, an ordered pair with first...

an ordered pair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a). While the two elements of an ordered pair (a...

axiom of pairing also allows for the definition of ordered pairs. For any objects a {\displaystyle a} and b {\displaystyle b} , the ordered pair is defined...

edges, directed links, directed lines, arrows, or arcs), which are ordered pairs of distinct vertices: E ⊆ { ( x , y ) ∣ ( x , y ) ∈ V 2 and x ≠ y }...

ordered pair (a pair ( x , y ) {\displaystyle (x,y)} which is the same type as its projections) in NFU. It is convenient to use the Kuratowski pair in...

Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. In terms...

<360^{\circ }} . Vectors can be specified using either ordered pair notation (a subset of ordered set notation using only two components), or matrix notation...

In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is...

Σ-types are more powerful than typical ordered pair types because of dependent typing. In the ordered pair, the type of the second term can depend on...

real algebras are as follows: The complex numbers can be written as ordered pairs (a, b) of real numbers a and b, with the addition operator being component-wise...

{ 1 , 2 } {\displaystyle \{1,2\}} ⁠, an indexed family is called an ordered pair. When the index set is the set of the ⁠ n {\displaystyle n} ⁠ first natural...

different contexts. The idempotents of B are all pairs (x, x), where x is any natural number (using the ordered pair characterisation of B). Since these commute...

ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x\in X} and y ∈ Y . {\displaystyle y\in Y.} The set of all these pairs is...

In one restricted but very common sense of the term, a graph is an ordered pair G = ( V , E ) {\displaystyle G=(V,E)} comprising: V {\displaystyle V}...

relation as an ordered pair using the null set. At approximately the same time, Hausdorff (1914, p. 32) gave the definition of the ordered pair (a, b) as {{a...

to the beginning of a list.) Although cons cells can be used to hold ordered pairs of data, they are more commonly used to construct more complex compound...

regularity enables defining the ordered pair (a,b) as {a,{a,b}}; see ordered pair for specifics. This definition eliminates one pair of braces from the canonical...

initial point to an end point; in this case, the bound vector is an ordered pair of points in the same position space, with all coordinates having the...

ISBN 978-0134689517. A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21...

has two arguments, x {\displaystyle x} and y {\displaystyle y} , in an ordered pair ( x , y ) {\displaystyle (x,y)} . The hypergeometric function is an example...

transitive meaning that any ordered pair of points can be mapped by at least one collineation to any other ordered pair of points. (See below.) Collineations...

{\displaystyle \equiv y} -axis (vertical) coordinate Together they form an ordered pair which defines the location of a point in two-dimensional rectangular...

something, a pair Unordered pair, or pair set, in mathematics and set theory Ordered pair, or 2-tuple, in mathematics and set theory Pairing, in mathematics...