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Pair of mathematical objects
In mathematics, an ordered pair, denoted (a, b), is a pair of objects in which their order is significant. If a and b are different, then (a,b) is different
Ordered_pair
Finite ordered list of elements
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)
Tuple
Mathematical set with an ordering
reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,≤) consisting of a set X (called the ground set
Partially_ordered_set
Number in {..., –2, –1, 0, 1, 2, ...}
{\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
Integer
Axiomatic set theory devised by W.V.O. Quine
It is straightforward that for any set x, x∪xc=V. Ordered Pair: For each a, b, the ordered pair of a and b, (a,b), exists; (a,b)=(c,d) exactly if a=c
New_Foundations
Use of coordinates for representing vectors
<360^{\circ }} . Vectors can be specified using either ordered pair notation (a subset of ordered set notation using only two components), or matrix notation
Vector_notation
Topics referred to by the same term
number theory Ordinate in mathematics, the y element of an ordered pair (x, y) Partially ordered set Complete partial order Permutation, the act of arranging
Order
Mathematical set formed from two given sets
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
Cartesian_product
Informal set theories
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
Naive_set_theory
Collection of mathematical objects
ai. When the index set is {1,2}, an indexed family is called an ordered pair. When the index set is the set of the n first natural numbers, an indexed
Set_(mathematics)
Vertices connected in pairs by edges
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 }
Graph_(discrete_mathematics)
Physical quantity that is a vector
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
Vector_quantity
Alternative foundation of mathematics
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
Intuitionistic_type_theory
Algebraic structure in linear algebra
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
Vector_space
Unordered set containing two elements
particular relation between them, where {a, b} = {b, a}. In contrast, an ordered pair (a, b) has a as its first element and b as its second element, which
Unordered_pair
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
Bicyclic_semigroup
the axioms of Pairing and Union (x∪y=⋃{x,y}). First, consider the ordered pair. The reason that this comes first is technical: ordered pairs are needed to
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Axiom of set theory
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
Axiom_of_regularity
Horizontal and vertical axes/coordinate numbers of a 2D coordinate system or graph
{\displaystyle \equiv y} -axis (vertical) coordinate Together they form an ordered pair which defines the location of a point in two-dimensional rectangular
Abscissa_and_ordinate
Method for producing composition algebras
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
Cayley–Dickson_construction
About mathematical functions
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
History of the function concept
History_of_the_function_concept
Concept in axiomatic set theory
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
Axiom_of_pairing
Topics referred to by the same term
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
Pair
Type of binary relation
vacuously true. A relation R containing only one ordered pair is also transitive: if the ordered pair is of the form ( x , x ) {\displaystyle (x,x)} for
Transitive_relation
Function and primitive data structure in Lisp and other functional programming languages
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
Cons
Geometric model of the planar projection of the physical universe
Cartesian coordinate system is called a Cartesian plane. The set ℝ2 of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product
Euclidean_plane
Geometry with 7 points and 7 lines
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
Fano_plane
Association of one output to each input
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
Function_(mathematics)
Order whose elements are all comparable
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
Total_order
Euclidean geometry without distance and angles
given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the
Affine_geometry
Relationship between two numbers of the same kind
is proportional to 3:4. Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the
Ratio
Square array with symbols that each occur once per row and column
ordered pairs (r, c) are distinct, all ordered pairs (r, s) are distinct, and all ordered pairs (c, s) are distinct. This means that the n2 ordered pairs
Latin_square
Type of mathematical array
orthogonal array with symbol set {1,2} and strength 2. Notice that the four ordered pairs (2-tuples) formed by the rows restricted to the first and third columns
Orthogonal_array
Graph with multiple edges between two vertices
loops. A multigraph G is an ordered pair G := (V, E) with V a set of vertices or nodes, E a multiset of unordered pairs of vertices, called edges or
Multigraph
2000 mathematics book by Lakoff & Núñez
n∪{n}. The ordered pair (a,b), defined as {{a},{a,b}}. By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both
Where_Mathematics_Comes_From
Sets with no element in common
instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the
Disjoint_sets
In mathematics, operation on sets
building the disjoint union is to define A {\displaystyle A} as the set of ordered pairs ( x , i ) {\displaystyle (x,i)} such that x ∈ A i , {\displaystyle x\in
Disjoint_union
Way of defining a lattice in the complex plane
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
Fundamental_pair_of_periods
Algebraic structure of set algebra
under complement, countable unions, and countable intersections. The ordered pair ( X , Σ ) {\displaystyle (X,\Sigma )} is called a measurable space. The
Σ-algebra
Relationship between elements of two sets
codomain. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x,y), where x is an element of X and y is an element of Y. It encodes
Binary_relation
Input to a mathematical function
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
Argument_of_a_function
3-volume treatise on mathematics, 1910–1913
"Relations" are what is known in contemporary set theory as sets of ordered pairs. Sections ✱20 and ✱22 introduce many of the symbols still in contemporary
Principia_Mathematica
Collection of objects, associated with an index set
positive integer. An ordered pair (2-tuple) is a family indexed by the set of two elements, 𝟐={1,2}; each element of the ordered pair is indexed by an element
Indexed_family
Mathematical concept
The ordered pairs ( 1 , 2 ) , ( 2 , 4 ) {\displaystyle (1,2),(2,4)} and ( 3 , 6 ) {\displaystyle (3,6)} serve as an example of this. These pairs belong
Equivalence_class
Type of binary relation
{\displaystyle bRa} , where the notation a R b {\displaystyle aRb} means that the ordered pair ( a , b ) {\displaystyle (a,b)} is in the relation R {\displaystyle R}
Symmetric_relation
Mathematical set containing no elements
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
Empty_set
Pair of positions in a sequence where two elements are out of sorted order
) {\displaystyle \pi (i)>\pi (j)} . The inversion is indicated by an ordered pair containing either the places ( i , j ) {\displaystyle (i,j)} or the elements
Inversion (discrete mathematics)
Inversion_(discrete_mathematics)
Algebraic structure
xy} , denotes the result of applying the semigroup operation to the ordered pair ( x , y ) {\displaystyle (x,y)} . Associativity is formally expressed
Semigroup
Symmetric arrangement of finite sets
the set of all ordered pairs consisting of the corresponding entries in the two squares has n2 distinct members (all possible ordered pairs occur). A set
Combinatorial_design
Arithmetic operation
vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as a vector from the origin in
Addition
Mathematical problem
superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
In the theory and practice of music, a fifth interval is an ordered pair of notes that are separated by an interval of 6–8 semitones. There are three types
List_of_fifth_intervals
theory, a discipline within mathematics, a critical pair is a pair of elements in a partially ordered set that are incomparable but that could be made comparable
Critical_pair_(order_theory)
Graph with oriented edges
directed graph is an ordered pair G = (V, A) where V is a set whose elements are called vertices, nodes, or points; A is a set of ordered pairs of vertices, called
Directed_graph
System of mathematical set theory
{\displaystyle \{A\}} exists. Given any two sets, their unordered and ordered pairs exist. Given any set of sets, its union exists. TG includes the following
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
Concept in algebraic topology
i'\circ g=f\circ i} . A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace. The use of pairs of spaces is sometimes
Topological_pair
couple is a class with exactly two elements 2. An ordinal couple is an ordered pair (treated in PM as a special sort of relation) Dedekindian complete (relation)
Glossary of Principia Mathematica
Glossary_of_Principia_Mathematica
is a symmetry (graph automorphism) taking any ordered pair of adjacent vertices to any other ordered pair; the Foster census lists all small symmetric
List_of_graphs
Topics referred to by the same term
+ 1 {\displaystyle j^{2}=+1} A 2-tuple, or ordered list of two elements, commonly called an ordered pair, denoted ( a , b ) {\displaystyle (a,b)} Double
Double
Set with exactly one element
set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A, the axiom applied to A and A asserts the existence of {
Singleton_(mathematics)
Use of braces for specifying sets
{\displaystyle \mathbb {R} \times \mathbb {R} } denotes the set of ordered pairs of real numbers. { n ∈ N ∣ ( ∃ k ) [ k ∈ N ∧ n = 2 k ] } {\displaystyle
Set-builder_notation
Concept in axiomatic set theory
{P}}(X\cup Y)} and, for example, considering a model using the Kuratowski ordered pair, ( x , y ) = { { x } , { x , y } } ∈ P ( P ( X ∪ Y ) ) {\displaystyle
Axiom_of_power_set
Extension of a concept, idea, or sign
exists). For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the
Extension_(semantics)
Algebra whose elements are stable matchings
the pairs from P {\displaystyle P} are part of this cyclic sequence, the pairs from Q {\displaystyle Q} consist of the second element of each ordered pair
Lattice_of_stable_matchings
All numbers between two given numbers
mathematics. For instance, the notation (a, b) is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry
Interval_(mathematics)
Hierarchy of formulas in set theory
infinite ordinal ω x is an ordered pair. The first entry of the ordered pair x is a. The second entry of the ordered pair x is b p. 14 f is a function
Lévy_hierarchy
Data storage paradigm in computing
An ordered key–value store (OKVS) is a type of data storage paradigm that can support multi-model databases. An OKVS is an ordered mapping of bytes to
Ordered_key–value_store
Mathematical operation
the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand;
Algebraic_operation
Condition required for a semantic statement to be true
way of representing the truth condition of "Nixon is alive" is as the ordered pair <Nixon, {x: x is alive}>. And we say that "Nixon is alive" is true if
Truth_condition
2. A Kuratowski ordered pair is a definition of an ordered pair using only set theoretical concepts, specifically, the ordered pair (a, b) is defined
Glossary_of_set_theory
Relationship between two sets, defined by a set of ordered pairs
are not. Formally, a relation R over a set X can be seen as a set of ordered pairs (x,y) of members of X. The relation R holds between x and y if (x,y)
Relation_(mathematics)
Type of logical system
that first-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projection functions. A first-order theory
First-order_logic
Mathematical function on ordinals
the second position in an ordered pair is never zero, i.e. a value of zero is indicated by the absence of an ordered pair; when using the set as a function
Veblen_function
Arithmetic function
_{k=1}^{n}\varphi (k),\quad n\in \mathbb {N} .} It is the number of ordered pairs of coprime integers (p,q), where 1 ≤ p ≤ q ≤ n. The first few values
Totient_summatory_function
Mathematical set that can be enumerated
generally: Theorem—A subset of a countable set is countable. The set of all ordered pairs of natural numbers (the Cartesian product of two sets of natural numbers
Countable_set
Form of geometry without distances
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion
Ordered_geometry
Graph in which all ordered pairs of linked nodes are automorphic
theory, a graph G is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices ( u 1 , v 1 ) {\displaystyle (u_{1},v_{1})} and
Symmetric_graph
Representation of a mathematical function
mathematics, the graph of a function f {\displaystyle f} is the set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle
Graph_of_a_function
Natural number
(ed.). "Sequence A071605 (Number of ordered pairs (a,b) of elements of the symmetric group S_n such that the pair a,b generates S_n)". The On-Line Encyclopedia
216_(number)
System of mathematical set theory
elements that are not ordered pairs, while the intersection E ∩ V 2 {\displaystyle E\cap V^{2}} contains only the ordered pairs of E {\displaystyle E}
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Mathematical tool for summing arithmetic functions
the inner sum runs over all ordered pairs (x,y) of positive integers such that xy = k. In the Cartesian plane, these pairs lie on a hyperbola, and when
Dirichlet_hyperbola_method
Abstract data type in computer science
with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also
Graph_(abstract_data_type)
Pair of logical equivalences
Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are
De_Morgan's_laws
Sporadic simple group
subgroups of M24, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate
Mathieu_group_M24
programming languages. The locks-and-keys approach represents pointers as ordered pairs (key, address) where the key is an integer value. Heap-dynamic variables
Locks-and-keys_(computing)
Mathematical concept
theorem of ZF, and relies on the fact that the ordinal numbers are well-ordered, and thus a statement that is not universally true for all ordinals must
Transfinite_induction
Basic concept in set theory
In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X,x0) where X is a set and x0 is an element of X called the base point
Pointed_set
Mathematical set of all subsets of a set
(z, 3) } is defined in which the number in each ordered pair represents the position of the paired element of S in a sequence of binary digits such as
Power_set
Fraction made by summing the numerator and denominator of two fractions
binary operation on valid fractions (nonzero denominator), considered as ordered pairs of appropriate integers, a priori disregarding the perspective on rational
Mediant_(mathematics)
System of mathematical set theory
z\leftrightarrow (s=x\,\lor \,s=y)])].} Pairing licenses the unordered pair in terms of which the ordered pair, ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle
Morse–Kelley_set_theory
= ⟨ x , x ⟩ {\displaystyle \delta _{A}(x)=\langle x,x\rangle } , the ordered pair formed from x {\displaystyle x} . The reason for the name is that the
Diagonal_morphism
S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if: For all u and v
Probabilistic_metric_space
Basic counting principle in mathematics
member of {A, B, C}, and then to do so again, in effect choosing an ordered pair each of whose components are in {A, B, C}, is 3 × 3 = 9. As another example
Rule_of_product
Number with a real and an imaginary part
Re(2+3i)=2, Im(2+3i)=3. A complex number z can be identified with the ordered pair of real numbers (ℜ(z),ℑ(z)), which may be interpreted as coordinates
Complex_number
Branch of mathematics
construction is the cartesian product of two partially ordered sets, taken together with the product order on pairs of elements. The ordering is defined by (a, x)
Order_theory
Collection of sets in mathematics that can be defined based on a property of its members
y = P ( x ) {\displaystyle y={\mathcal {P}}(x)} . The fact that the ordered pair ( x , y ) {\displaystyle (x,y)} satisfies Φ {\displaystyle \Phi } may
Class_(set_theory)
Data representation method in computing systems
A name–value pair, also known as an attribute–value pair, key–value pair, or field–value pair, is a fundamental data representation in computer systems
Name–value_pair
Fundamental unit of which graphs are formed
set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram
Vertex_(graph_theory)
Concept in group theory
distinct ordered pairs {(x,y)∈S×S:x≠y}. That is, assuming (without a real loss of generality) that G acts on the left of S, for each pair of pairs (x,y)
Multiply transitive group action
Multiply_transitive_group_action
Set whose elements all belong to another set
order, in the sense that every partially ordered set (X,⪯) is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple
Subset
travel, tourism, insurance
ORDERED PAIR
ORDERED PAIR
Girl/Female
English, Peruvian
Plaster; Powdered
Girl/Female
African, Arabic, Muslim
Well-ordered; Well-arranged
Boy/Male
Hindu, Indian, Telugu
Bordered; Friendly Element
Boy/Male
Hindu
Orderly
Girl/Female
Greek
Murdered Agamemnon.
Male
English
Old English Arthurian legend name of a Knight of the Round Table who was the illegitimate son and traitor of King Arthur, possibly MORDRED means "sea counsel." He was brother (or half-brother) to Agravain, Gaheris, Gareth, and Gawain, and noted for having crowned himself and married Guinevere while Arthur was waging war on Emperor Lucius of Rome. He was killed by Arthur at the Battle of Camlann.Â
Boy/Male
Arabic, Australian, Muslim
Ordered; Appointed
Girl/Female
Shakespearean
The Tragedy of Macbeth' Lady Macduff, wife to Macduff, murdered on Macbeth's orders.
Girl/Female
Muslim
Well-arranged, Well-ordered
Boy/Male
Tamil
Mitanshu | மீதாஂஷà¯Â
Bordered, Friendly element
Mitanshu | மீதாஂஷà¯Â
Boy/Male
American, British, Christian, English
Brave; Brave Counselor
Boy/Male
English Arthurian Legend
Brave.
Boy/Male
Indian
Responsibility; Ordered
Boy/Male
Indian
Ordered, Pasted, Appointed
Boy/Male
African, Indian, Sanskrit
Clear Spoken Person; Ordered
Boy/Male
Tamil
Orderly
Male
Arthurian
, a son of Lot; traitor to Arthur.
Girl/Female
Indian
Well-arranged, Well-ordered
Boy/Male
Muslim
Ordered, Pasted, Appointed
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a place in Lancashire, called Ormerod, from the Old Norse personal name Ormr (see Orme 1) or Ormarr (a compound of orm ‘serpent’ + herr ‘army’) + Old English rod ‘clearing’.
ORDERED PAIR
ORDERED PAIR
ORDERED PAIR
ORDERED PAIR
ORDERED PAIR
ORDERED PAIR
ORDERED PAIR
a.
Having three prominent longitudinal angles; as, a three-cornered stem.
a.
Observant of order, authority, or rule; hence, obedient; quiet; peaceable; not unruly; as, orderly children; an orderly community.
n.
One who puts in order, arranges, methodizes, or regulates.
n.
To give an order to; to command; as, to order troops to advance.
a.
Covered or adorned with osiers; as, osiered banks.
n.
To admit to holy orders; to ordain; to receive into the ranks of the ministry.
n.
Right arrangement; a normal, correct, or fit condition; as, the house is in order; the machinery is out of order.
n.
An assemblage of genera having certain important characters in common; as, the Carnivora and Insectivora are orders of Mammalia.
a.
Well-ordered; orderly; regular; methodical.
a.
Being on duty; keeping order; conveying orders.
a.
Conformed to order; in order; regular; as, an orderly course or plan.
v. i.
To give orders; to issue commands.
n.
One who gives orders.
a.
Having three corners, or angles; as, a three-cornered hat.
a.
Performed in good or established order; well-regulated.
adv.
According to due order; regularly; methodically; duly.
n.
An ecclesiastical grade or rank, as of deacon, priest, or bishop; the office of the Christian ministry; -- often used in the plural; as, to take orders, or to take holy orders, that is, to enter some grade of the ministry.
n.
A noncommissioned officer or soldier who attends a superior officer to carry his orders, or to render other service.
imp. & p. p.
of Order
n.
To give an order for; to secure by an order; as, to order a carriage; to order groceries.
travel, tourism, insurance