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Unordered set containing two elements
In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them
Unordered_pair
Pair of mathematical objects
different from (b,a). In contrast, the unordered pair {a,b} always equals the unordered pair {b,a}. Ordered pairs are also called 2-tuples, or sequences
Ordered_pair
Concept in axiomatic set theory
of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced
Axiom_of_pairing
Vertices connected in pairs by edges
multigraph) is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {v1,v2} of vertices
Graph_(discrete_mathematics)
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
stratified comprehension; in ZFC, the existence of the unordered pair is given by the Axiom of Pairing, the existence of the empty set follows by Separation
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Axiom of set theory
for any (even infinite) collection of unordered pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate collection (i
Axiom_of_choice
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
Geometry with 7 points and 7 lines
there are 21 unordered pairs of points, each of which may be mapped by a symmetry onto any other unordered pair. For any unordered pair there are 8 symmetries
Fano_plane
Abstract data type in computer science
together 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
Graph_(abstract_data_type)
Fundamental unit of which graphs are formed
a 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
Vertex_(graph_theory)
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 lines
Multigraph
contains an unordered pair from the set of symbols Each symbol occurs exactly once in each row and column of the array Every unordered pair of symbols
Room_square
Type of triangular number
triangular numbers arise naturally as numbers of unordered pairs of unordered pairs of objects, including pairs where both objects are the same: An example
Doubly_triangular_number
Finite ordered list of elements
types, featuring unordered elements accessed by label. A few programming languages combine ordered tuple product types and unordered record types into
Tuple
Strongly regular graph
graph correspond to the 231 unordered pairs of elements. Two vertices are adjacent whenever they come from two disjoint pairs whose union belongs to one
Cameron_graph
Expression in mathematics
ISBN 978-0-321-39053-0. Dijkstra, Edsger W. (1997). "WLOG, or the misery of the unordered pair (EWD1223)". In Broy, Manfred; Schieder, Birgit (eds.). Mathematical
Without_loss_of_generality
System of mathematical set theory
A } {\displaystyle \{A\}} exists. Given any two sets, their unordered and ordered pairs exist. Given any set of sets, its union exists. TG includes the
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
Special points within a triangle
a similarity, turns one Brocard point into the other. However, the unordered pair formed by both points is invariant under similarities. The midpoint
Brocard_points
Graph with oriented edges
ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, links or lines. The aforementioned
Directed_graph
usually written ∀ unordered pair A set of two elements where the order of the elements does not matter, distinguishing it from an ordered pair where the sequence
Glossary_of_set_theory
Kind of large cardinal number
that is, either f maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one. An equivalent formulation
Ineffable_cardinal
Concept in mathematics
names of the points". The unordered configuration space is sometimes denoted 𝒰𝒞n(X), Bn(X), or Cn(X). The collection of unordered configuration spaces over
Configuration space (mathematics)
Configuration_space_(mathematics)
Symmetric arrangement of finite sets
empty or contains an unordered pair from S, Each symbol occurs exactly once in each row and column of the array, and Every unordered pair of symbols occurs
Combinatorial_design
Sporadic simple group
automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points. M11 has a 3-transitive
Mathieu_group_M11
Measure of similarity between two data clusterings
probability that X and Y will agree on a randomly chosen pair. The number of all possible unordered pairs, the binomial coefficient (n2), can be calculated as
Rand_index
Formula concerning prime numbers
congruent to a can be grouped into unordered pairs (x,y) according to the rule that the product of the members of each pair is congruent to a modulo p (since
Euler's_criterion
Non-orientable surface with one edge
the hyperbolic plane can be parameterized by unordered pairs of distinct points on a circle, the pairs of points at infinity of each line. This space
Möbius_strip
2000 mathematics book by Lakoff & Núñez
pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999)
Where_Mathematics_Comes_From
Data structure representing a graph
science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the
Adjacency_list
System of mathematical set theory
licenses the unordered pair in terms of which the ordered pair, ⟨x,y⟩, may be defined in the usual way, as {{x},{x,y}}. With ordered pairs in hand, Class
Morse–Kelley_set_theory
Every graph has evenly many odd vertices
undirected graph consists of a system of vertices, and edges connecting unordered pairs of vertices. In any graph, the degree deg ( v ) {\displaystyle \deg(v)}
Handshaking_lemma
American inventor of algorithmic probability and artificial intelligence researcher
Distribution and associated convergence theorems to unordered sets of strings and a 2008 report, to unordered pairs of strings. In 1997, 2003 and 2006 he showed
Ray_Solomonoff
Associative array for storing key–value pairs
table structures, such as Python’s dictionaries, Java’s HashMap, C++’s unordered_map, and Go maps, which abstract the complexity of hashing from the programmer
Hash_table
Polynomial in combinatorial mathematics
{1}{4}}\left(a_{1}^{4}+a_{2}^{2}+2a_{4}\right).} The group C4 also acts on the unordered pairs of elements of X in a natural way. Any permutation g would send {x
Cycle_index
Generalization of the one-dimensional normal distribution to higher dimensions
the sum is taken over all allocations of the set {1,…,2λ} into λ (unordered) pairs. That is, for a kth (= 2λ = 6) central moment, one sums the products
Multivariate normal distribution
Multivariate_normal_distribution
Algorithm in graph theory
graph) G is a pair of sets (V, E) where V is a nonempty set called the set of vertices of G, and E is a (possibly empty) set of unordered pairs of distinct
Havel–Hakimi_algorithm
Polynomial sequence
coefficient of xk is the number of (unordered) partitions of an n-element set into k singletons and n − k/2 (unordered) pairs. Equivalently, it is the number
Hermite_polynomials
Metric to compare ordering
P,i<j}{\bar {K}}_{i,j}(\tau _{1},\tau _{2})} where P is the set of unordered pairs of distinct elements in τ 1 {\displaystyle \tau _{1}} and τ 2 {\displaystyle
Kendall_tau_distance
Homotopic map of a graph
representative of φ. Let f : Γ → Γ be a combinatorial map. A turn is an unordered pair e, h of oriented edges of Γ (not necessarily distinct) having a common
Train_track_map
System of mathematical set theory
all ordered pairs (a, b) of elements a of A and b of B. Proof: The singleton set with member a, written {a}, is the same as the unordered pair {a, a}, by
Kripke–Platek_set_theory
and the remaining (t2) of which can be labeled by unordered pairs of these integers. For each pair of integers i,j in this range, it has a hyperedge whose
Hedgehog_(hypergraph)
Associative data types in the C++ standard library
key–value pairs, while a set stores keys alone. The multi variants allow more than one element with an equivalent key. The standard library's unordered
Associative_containers_(C++)
Sporadic simple group
has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M21.2, and the other
Mathieu_group_M23
2 points about which a triangle can be inverted into an equilateral triangle
transformations: the unordered pair of isodynamic points of a transformation of ABC is equal to the same transformation applied to the pair {S,S′}. The individual
Isodynamic_point
Representation of a graph's triconnected components
the SPQR tree plus, for every S-node with k vertices, the number of unordered pairs of non-adjacent vertices in the corresponding cycle (i.e., k(k − 3)/2)
SPQR_tree
System for displaying information by using different colors
– the colors are unordered, but are chosen to maximize saliency of the colors, by maximizing color difference between all color pair permutations. Continuous
Color_code
Non-convex polyhedron with no triangulation
eight triangular faces. Its six vertices form fifteen unordered pairs. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in
Schönhardt_polyhedron
Minimum monochromatic-triangle theorem in graph theory
counts unordered pairs of differently colored edges incident with v {\displaystyle v} . Every nonmonochromatic triangle contributes such a pair at exactly
Goodman's_theorem
Set of unordered triples from a vertex set
collection of unordered triples is a two-graph: 123 124 135 146 156 236 245 256 345 346 This two-graph is a regular two-graph since each pair of distinct
Two-graph
formal setting, a network g {\displaystyle g} is defined as a set of unordered pairs { i , j {\displaystyle i,j} }, with i , j {\displaystyle i,j} element
Strategic_network_formation
( 6 2 ) {\displaystyle 15={\tbinom {6}{2}}} unordered pairs of elements of a six-element set; these pairs are called duads. Similarly, the lines of the
Cremona–Richmond configuration
Cremona–Richmond_configuration
Mathematical theory
extensionality axiom"; axioms asserting the existence of the empty set, unordered pair, union set, and power set; the axiom of separation; an axiom stating
Quasi-set_theory
Choice of colors used in design
schemes. Qualitative and quantitative color schemes are used to encode unordered categorical data and ordered data, respectively. Color schemes are often
Color_scheme
Task of computing complete subgraphs
undirected graph is formed by a finite set of vertices and a set of unordered pairs of vertices, which are called edges. By convention, in algorithm analysis
Clique_problem
objects of the same type, based on user-defined operators == and <. std::pair<T, U>, a container template which holds two member objects (first and second)
Utility_(C++)
Numbers parameterizing ways to partition a set
ordered pairs of subsets remain. Finally, since we want unordered pairs rather than ordered pairs we divide this last number by 2, giving the result above
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Distance between unordered pitch classes
set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even
Interval_class
Invariant in projective geometry
24 ways, but there are only six ways for partitioning them into two unordered pairs. Thus, four points can have only six different cross-ratios, which
Cross-ratio
2011 edition of the C++ programming language standard
of the header <functional> and the introduction of headers <unordered_set> and <unordered_map>. No other changes to any existing standard classes were
C++11
Document that proposed additions to the C++ standard library
std::vector new <unordered_set>, <unordered_map> header files they implement the unordered_set, unordered_multiset, unordered_map, and unordered_multimap classes
C++_Technical_Report_1
Software library for the C++ programming language
steps in one operation). A fundamental STL concept is a range which is a pair of iterators that designate the beginning and end of the computation, and
Standard_Template_Library
of edges that are unordered pairs of vertices. The open neighborhood of any vertex is the set of other vertices that it is paired with in edges of the
Twin-width
Punctuation mark
parentheses are used to denote ordered pairs and other tuples, whereas curly brackets are used for unordered sets. In physical sciences and statistical
Bracket
Concept in mathematics
construction is that each of the levels is not closed under the formation of unordered pairs; for a given x , y ∈ L α + 1 ∖ L α {\displaystyle x,y\in L_{\alpha
Jensen_hierarchy
American mathematician
the theory of ordinal definable sets, an unordered pair of sets is said to be a Groszek–Laver pair if the pair is ordinal definable but neither of its
Marcia_Groszek
vertex set | A | {\displaystyle |{\mathcal {A}}|} . Its edges are the unordered pairs { α , β } {\displaystyle \{\alpha ,\beta \}} such that α ∼ β {\displaystyle
Coherent_space
Probability inequality for dependent events
between them: it adds up, over all unordered pairs of distinct sets that overlap, the probability that both occur. Pairs of sets that do not overlap depend
Janson_inequality
Generic type features in C++
std::equal_to<K>, typename Alloc = std::allocator<std::pair<const K, T>> > using HashMap = std::unordered_map<K, V, HashFn, KeyEq, Alloc>; // or, only allowing
Template_(C++)
{\displaystyle H=(V(G),E(G))} is a clutter (if edges are treated as unordered pairs of vertices) and b ( H ) {\displaystyle b(H)} is the collection of
Sperner_family
partial order over the events in the trace is constructed, and any unordered pairs of critical events are reported as races. Many predictive techniques
Runtime_predictive_analysis
Collection of objects, associated with an index set
i-th vector vi only makes sense with respect to this family, as sets are unordered so there is no i-th vector of a set. Furthermore, linear independence
Indexed_family
Layer for end-to-end network communication
address with a transport-layer port; in TCP, a connection is identified by a pair of sockets. TCP provides a connection-oriented, reliable, in-order byte-stream
Transport_layer
Theorem in projective geometry
lie on a common line, the last point will be on that line, too. If six unordered points are given on a conic section, they can be connected into a hexagon
Pascal's_theorem
Natural number
). "Sequence A000954 (Conjecturally largest even integer which is an unordered sum of two primes in exactly n ways)". The On-Line Encyclopedia of Integer
68_(number)
Revision of the C++ programming language planned for 2029
std::map<K, V>, std::unordered_map<K, V> and std::flat_map<K, V> which returns a std::optional<V&> A tagged pointer class std::pointer_tag_pair<Ptr, Tag, N> with
C++29
Data structure holding key/value pairs
a normal dictionary, or by moving the actual data out of the sparse (unordered) array and into a dense insertion-ordered one. Associative arrays can
Associative_array
Mathematical function
values between a pair may be larger than it, one may also consider the permutations of this multiset in which the first copies of each pair appear in sorted
Double_factorial
Method for encoding data in communications
have been introduced by Donald Knuth; they are a subset of so-called unordered codes, which are codes having the property that the positions of ones
Constant-weight_code
Performance-limiting phenomenon in computer networks
Wolinsky, David; Ford, Bryan (2013). Reducing Latency in Tor Circuits with Unordered Delivery. 3rd USENIX Workshop on Free and Open Communications on the Internet
Head-of-line_blocking
Descriptive statistic
setting where an intraclass correlation is desired, the pairs are considered to be unordered. For example, if we are studying the resemblance of twins
Intraclass_correlation
pairs can also be applied to triples and quads. X-wing – See N-fish (with N=2). Swordfish – See N-fish (with N=3). N-fish – Analogues of hidden pairs/triples/quads
Glossary_of_sudoku
Computing project for a user locale data format
XML file and interpret it as an unordered list of <path,value> pairs, called a CLDRFile internally. These path/value pairs can be added to or deleted, and
Common_Locale_Data_&_Rules
2006 film by Alfonso Cuarón
Foundas, The Village Voice Unordered – Dana Stevens, Slate Unordered – Liam Lacey and Rick Groen, The Globe and Mail Unordered – Peter Rainer, The Christian
Children_of_Men
Graphical tool in probability
constraints explained as follows. Recall that an edge in a tree is an unordered set of two nodes. Each edge in a vine is associated with a constraint
Vine_copula
Abstract data type for storing distinct values
search tree. C++11 introduces the std::unordered_multiset class for the unsorted multiset, as a kind of unordered associative container, which implements
Set_(abstract_data_type)
Abstract data type in computer science
a set of n elements. Variants of the basic heap data structure such as pairing heaps or Fibonacci heaps can provide better bounds for some operations
Priority_queue
Individual component of an HTML document
1.2, and was standardized in HTML 2.0; still current. <ul>...</ul> An unordered (bulleted) list. The type of list item marker can be specified in an HTML
HTML_element
different spin. For example, a "jump topspin" and a "hybrid" serve are often paired at high levels to deceive passers Jump float : A serve in which the player
Volleyball_jargon
Algorithm that arranges lists in order
worst-case performance is O(n2), so it is rarely used to sort large, unordered data sets. Bubble sort can be used to sort a small number of items (where
Sorting_algorithm
Difference in pitch between two notes
distance between two pitches without taking into account direction with the unordered pitch interval, somewhat similar to the interval of tonal theory. The
Interval_(music)
Graphical model of computation
left-hand side is called an interface, while the right-hand side contains an unordered multiset of equations v i = w i {\displaystyle v_{i}=w_{i}} . Wiring ω
Interaction_nets
Computer programming method for hashing
Mark; Kuszmaul, William (2024-10-27), "Tight Analyses of Ordered and Unordered Linear Probing", 2024 IEEE 65th Annual Symposium on Foundations of Computer
Linear_probing
Commonly-accepted chronology of early Roman history
by the various chronologies to consular colleges should be taken as an unordered numeric identifier for those colleges rather than absolute years. The
Varronian_chronology
Number of subsets of a given size
usually read as "n choose k" because there are (nk) ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there
Binomial_coefficient
General-purpose programming language
slightly different semantics. For example, an alternative may include unordered dictionaries, in contrast to other current Python versions. As another
Python_(programming_language)
Mathematical concept for comparing objects
same length as u and that dominates it term by term. When (X,=) is an unordered set, u≤v if and only if u is a subsequence of v. (Xω,≤), the set of infinite
Well-quasi-ordering
1983 album by Charly García
Working from a set of songs García had already demoed in Argentina, the pair began rerecording and reshaping the material. García later recalled that
Clics_modernos
1971 song by Led Zeppelin
Plant Unledded. Najma Akhtar [de] sang Denny's vocal part. (*) designates unordered lists. List of cover versions of Led Zeppelin songs § The Battle of Evermore
The_Battle_of_Evermore
C++ Standard Library header providing algorithm implementations
algorithms defined in the <algorithm> header which operate on ranges rather than pairs of iterators. The ranges versions of algorithm functions are scoped within
Algorithm_(C++)
travel, tourism, insurance
UNORDERED PAIR
UNORDERED PAIR
Girl/Female
Muslim
First born of a pair
Boy/Male
Tamil
Mitanshu | மீதாஂஷà¯Â
Bordered, Friendly element
Mitanshu | மீதாஂஷà¯Â
Surname or Lastname
English
English : habitational name from a pair of villages in Cheshire, on either side of the Weaver river, recorded in Domesday Book as Maneshale, from the genitive case of the Old English personal name Mann + Old English scylf ‘shelf’, ‘ledge’.
Girl/Female
Muslim
First born of a pair
Boy/Male
Indian
Ordered, Pasted, Appointed
Boy/Male
Arabic, Australian, Muslim
Ordered; Appointed
Surname or Lastname
English (Lancashire)
English (Lancashire) : probably ‘brother of someone called Fair’ or else a descriptive name for the better-looking of a pair of brothers.
Boy/Male
Muslim
Ordered, Pasted, Appointed
Girl/Female
Muslim
Well-arranged, Well-ordered
Boy/Male
Hindu, Indian, Telugu
Bordered; Friendly Element
Male
Greek
(ἩÏώδης) Greek name HERODES means "sprung from a hero." In the bible, this is the name of the king who ordered the slaughter of all male children "two years old and under."Â
Girl/Female
Indian
First born of a pair
Boy/Male
Tamil
A pair, A month of kerala midhunam
Girl/Female
African, Arabic, Muslim
Well-ordered; Well-arranged
Surname or Lastname
English
English : habitational name from any of the numerous places so called. The final syllable represents Old English tÅ«n ‘enclosure’, ‘settlement’. The first element has a wide variety of possible origins. In the case of three examples in Lincolnshire it is Old English hÅh ‘spur of a hill’; for places in Oxfordshire and Somerset it is Old English halh ‘nook’, ‘recess’; for one in Dorset it may be Old English holh ‘hollow’, ‘depression’ or holt ‘small wood’; for a further pair in Suffolk it may be hola, genitive plural of holh ‘hollow’, but more probably a personal name HÅla.
Boy/Male
Indian
Responsibility; Ordered
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Boy/Male
African, Indian, Sanskrit
Clear Spoken Person; Ordered
Male
Greek
(ΑγÏίππας) Greek name AGRIPPAS means "wild horse." In the New Testament bible, this is the name of the Herod Agrippa who ordered the execution of the apostle James, and the imprisonment of Peter.
Girl/Female
Indian
Well-arranged, Well-ordered
UNORDERED PAIR
UNORDERED PAIR
UNORDERED PAIR
UNORDERED PAIR
UNORDERED PAIR
UNORDERED PAIR
UNORDERED PAIR
travel, tourism, insurance