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Mathematical concept for comparing objects
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments
Equivalence_relation
Mathematical concept for comparing objects
mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous
Partial_equivalence_relation
Mathematical concept
a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle S} into equivalence classes. These
Equivalence_class
Binary relation over a set and itself
strict chain, is a relation that is irreflexive, antisymmetric, transitive and connected. A partial equivalence relation is a relation that is symmetric
Homogeneous_relation
Type of binary relation
a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among
Transitive_relation
areas of mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that can
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
Relationship between elements of two sets
of homogeneous relations, a partial equivalence relation is difunctional. A strict order on a set is a homogeneous relation arising in order theory. In
Binary_relation
Reflexive and transitive binary relation
cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought
Preorder
Relationship between two sets, defined by a set of ordered pairs
of properties. A partial order is a relation that is reflexive, antisymmetric, and transitive, an equivalence relation is a relation that is reflexive
Relation_(mathematics)
Mathematical ways to group elements of a set
one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with
Partition_of_a_set
Mathematical construction of a set with an equivalence relation
mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional
Setoid
Topics referred to by the same term
efficiency rating, a measure of basketball player performance Partial equivalence relation, class of relations that are symmetric and transitive Perseus
Per
Theorem in numerical analysis
Lax equivalence theorem is a fundamental theorem in the analysis of linear finite difference methods for the numerical solution of linear partial differential
Lax_equivalence_theorem
Binary relation that relates every element to itself
symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. The word reflexive is originally derived from the Medieval
Reflexive_relation
Type of binary relation
reflexivity and transitivity, are the three defining properties of an equivalence relation. "is equal to" (equality) (whereas "is less than" is not symmetric)
Symmetric_relation
Mathematical set with an ordering
definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality. Wallis
Partially_ordered_set
Reversal of the order of elements of a binary relation
(weak order), or an equivalence relation, its converse is too. If I {\displaystyle I} represents the identity relation, then a relation R {\displaystyle
Converse_relation
Abstract mathematics relationship
category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories
Equivalence_of_categories
Mathematical ranking of a set
strict partial order < {\displaystyle \,<\,} is a strict weak order if and only if its induced incomparability relation is an equivalence relation. In this
Weak_ordering
Isometric subgraph of a hypercube
in the graph. Given a partial cube, it is straightforward to construct the equivalence classes of the Djoković–Winkler relation by doing a breadth first
Partial_cube
Differential geometry technique
In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up
Cartan's_equivalence_method
Relation of degree three
indeed this binary relation has some natural properties, like being an equivalence relation; while the combined ternary relation in general is not studied
Ternary_relation
generally, Quillen defines homotopy classes using the equivalence relation generated by the homotopy relation. A map K → L {\displaystyle K\to L} between Kan
Simplicial_homotopy
Physical law for definition of temperature
formulation of thermodynamics. It makes the relation of thermal equilibrium between systems an equivalence relation, which can represent equality of some quantity
Zeroth_law_of_thermodynamics
Mathematical relation inside orderings
covering relation is commonly used to graphically express the partial order by means of the Hasse diagram. Let X {\displaystyle X} be a set with a partial order
Covering_relation
Type of binary relation
pay each other's bills, the relation is antisymmetric. Partial and total orders are antisymmetric by definition. A relation can be both symmetric and antisymmetric
Antisymmetric_relation
to be W*W resp. WW*. A pair of projections are partitioned by the equivalence relation: P = W ∗ W , Q = W W ∗ {\displaystyle P=W^{*}W,\,Q=WW^{*}} It plays
Partial_isometry
Operation on the subsets of a set
of a relation is the smallest preorder containing it. Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the
Closure_(mathematics)
Branch of mathematics
also the only relation that is both a partial order and an equivalence relation because it satisfies both the antisymmetry property of partial orders and
Order_theory
Overview of and topical guide to logic
relation Directed set Equivalence relation Euclidean relation Homogeneous relation Idempotence Intransitivity Involutive relation Partial equivalence
Outline_of_logic
Property of a relation on a set
only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected
Connected_relation
Type of differential equation
. Jost, J. (2002), Partial Differential Equations, New York: Springer-Verlag, ISBN 0-387-95428-7. Olver, P.J. (1995), Equivalence, Invariants and Symmetry
Partial_differential_equation
Category where every morphism is invertible; generalization of a group
g:A\rightarrow C} . Special cases include: Setoid: a set that comes with an equivalence relation, G-set: a set equipped with an action of a group G {\displaystyle
Groupoid
Mathematical concept for comparing objects
than partial orders... it is simply more convenient to do so. Observe that a wpo is a wqo, and that a wqo gives rise to a wpo between equivalence classes
Well-quasi-ordering
Topological space construction
mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set
Quotient_space_(topology)
Marker used in SQL databases to indicate a value does not exist
always return UNKNOWN as the result of the expression. This is a partial equivalence relation and makes SQL an example of a Non-Reflexive logic. Similarly
Null_(SQL)
Numerical ordering with a margin of error
Therefore, the binary relation ( X , ≤ ) {\displaystyle (X,\leq )} defined in this way meets the three requirements of a partial order that it be reflexive
Semiorder
nested parentheses. 2. Equivalence class: given an equivalence relation, [ x ] {\displaystyle [x]} often denotes the equivalence class of the element x
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Type of binary relation
number n. In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order
Well-founded_relation
On chains and antichains in partial orders
chains needed to cover all elements. This number is called the width of the partial order. The theorem is named for the mathematician Robert P. Dilworth, who
Dilworth's_theorem
Theories in mathematical logic
transcendental. The signature of equivalence relations has one binary infix relation symbol ~, no constants, and no functions. Equivalence relations satisfy the
List_of_first-order_theories
Mathematical concept in category theory
{\text{and}}\ v:T\to A} with codomain A {\displaystyle A} , we define an equivalence relation by u ≡ v {\displaystyle u\equiv v} if there exists an isomorphism
Subobject
Partial order with joins
necessarily monotone with respect to the associated ordering relation. There is a well-known equivalence between the category S {\displaystyle {\mathcal {S}}}
Semilattice
Ways how entities stand to each other
types of relations, like equivalence and strict partial order. An equivalence relation is a relation that is reflexive, symmetric, and transitive, like
Relation_(philosophy)
Mathematical proposition equivalent to the axiom of choice
word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be comparable under the order relation, that
Zorn's_lemma
Informal use of mathematical notation
classes of non-Abelian groups of order 8". Referring to an equivalence class of an equivalence relation by x {\displaystyle x} instead of [ x ] {\displaystyle
Abuse_of_notation
Generalization of strings in computer science
independency relation. These induce an equivalence relation of equivalent strings; the elements of the equivalence classes are the traces. The equivalence relation
Trace_monoid
Equivalence of partially ordered sets
characteristics of an equivalence relation: reflexivity, symmetry, and transitivity. Therefore, order isomorphism is an equivalence relation. The class of partially
Order_isomorphism
One-to-one correspondence
is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a
Bijection
Continuous deformation between two continuous functions
above. Being homotopic is an equivalence relation on the set of all continuous functions from X to Y. This homotopy relation is compatible with function
Homotopy
Isomorphism type of ordered sets
different orders. Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes. If a set X {\displaystyle
Order_type
Type of logical relation
total relation. On the other hand, if f is a partial function, then the domain may be a proper subset of X, in which case f is not a total relation. "A
Total_relation
Order whose elements are all comparable
order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on
Total_order
Square array with symbols that each occur once per row and column
are isotopic are, in fact, equal. Isomorphism is also an equivalence relation and its equivalence classes are called isomorphism classes. Another type of
Latin_square
Visual depiction of a partially ordered set
endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. Hasse diagrams are named after Helmut Hasse (1898–1979); according
Hasse_diagram
Term in the mathematical area of order theory
isomorphism. Since partial orders are antisymmetric, the only ones that are self-dual are the equivalence relations (but the notion of partial order is self-dual)
Duality_(order_theory)
Concept in topology
points is an equivalence relation. No matter what topological space X might be to begin with, the quotient space under this equivalence relation is always
Kolmogorov_space
Language consisting of balanced strings of brackets
clear from the definition. The equivalence relation partitions the language Σ ∗ {\displaystyle \Sigma ^{*}} into equivalence classes. If we take ϵ {\displaystyle
Dyck_language
Description of limiting behavior of a function
}{\frac {f(x)}{g(x)}}=1.} The symbol ~ is the tilde. The relation is an equivalence relation on the set of functions of x; the functions f and g are said
Asymptotic_analysis
Concept in order theory
only if x ∧ y = x . {\displaystyle x\wedge y=x.} In fact, this relation is a partial order on A . {\displaystyle A.} Indeed, for any elements x , y
Join_and_meet
A binary relation R {\displaystyle R} is: An equivalence relation if R {\displaystyle R} is reflexive, symmetric, and transitive. A partial order if R
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Function, homomorphism, or morphism
R\ {\text{for some}}\ b\in B\}.} A relation ϕ ⊆ A × B {\displaystyle \phi \subseteq A\times B} is called a partial mapping of A to B if < a , b >∈ ϕ
Map_(mathematics)
Additional mathematical object
orders, graphs, events, differential structures, categories, setoids, and equivalence relations. Sometimes, a set is endowed with more than one feature simultaneously
Mathematical_structure
Property of elements related by inequalities
short descriptions of redirect targets, a partial ordering in which incomparability is a transitive relation Trotter, William T. (1992), Combinatorics
Comparability
Mathematical property
Generally, a preorder leads to an equivalence relation and a partial order between the corresponding equivalence classes. The Schröder–Bernstein property claims
Schröder–Bernstein_property
Two interrelated physics theories by Albert Einstein
of gravity. General relativity explains the law of gravitation and its relation to the forces of nature. It applies to the cosmological and astrophysical
Theory_of_relativity
Concept in model theory
observation permits one to define an equivalence relation among structures, reminiscent of the homotopy equivalence among topological spaces. Two structures
Interpretation_(model_theory)
Form of continuity for functions
said to be equivalent. Thus absolute continuity induces a partial ordering of such equivalence classes. If μ {\displaystyle \mu } is a signed or complex
Absolute_continuity
intersection is H: a H b if and only if a L b and a R b. This is also an equivalence relation on S. The class Ha is the intersection of La and Ra. More generally
Green's_relations
Operation in differential geometry
defined to be the set of equivalence classes of C p ∞ ( M , N ) {\displaystyle C_{p}^{\infty }(M,N)} modulo the equivalence relation E p k {\displaystyle
Jet_(mathematics)
Software testing technique that tests boundary values
on that set. Those inputs which belong to the same equivalence class as defined by the equivalence partitioning theory would constitute the basis. Given
Boundary-value_analysis
Type of residuated Boolean algebra with extra structure
any equivalence relation on the set X {\displaystyle X} . This is a generalization because X 2 {\displaystyle X^{2}} is itself an equivalence relation, namely
Relation_algebra
Binary relation which never occurs in both directions
x.} More generally, any strict partial order is an asymmetric relation. Not all asymmetric relations are strict partial orders. An example of an asymmetric
Asymmetric_relation
Topological spaces whose union is a boundary
{\displaystyle N} , ∂ W = M ⊔ N {\displaystyle \partial W=M\sqcup N} . Cobordisms are studied both for the equivalence relation that they generate, and as objects
Cobordism
Generalization of the real numbers
to the order relation ≤ given by the comparison rule below. The numeric forms are placed in equivalence classes; each such equivalence class is a surreal
Surreal_number
Mathematical logic concept
{\displaystyle (A\to B)\leftrightarrow (\neg B\to \neg A).} In practice, this equivalence can be used to make proving a statement easier. For example, if one wishes
Contraposition
Set whose elements all belong to another set
{\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal
Subset
Assignment of vector fields to manifolds
This defines an equivalence relation on the set of all differentiable curves initialized at x {\displaystyle x} , and equivalence classes of such curves
Tangent_space
In mathematics, invertible homomorphism
trichotomous, a partial order, total order, well-order, strict weak order, total preorder (weak order), an equivalence relation, or a relation with any other
Isomorphism
mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition
Series-parallel_partial_order
Set that intersects every one of a family of sets
general, since any equivalence relation on an arbitrary set gives rise to a partition, picking any representative from each equivalence class results in
Transversal_(combinatorics)
Algebraic structure
to the quotient of S by the equivalence relation ~ such that x ~ y if and only if f(x) = f(y). This equivalence relation is a semigroup congruence, as
Semigroup
Area of mathematical logic
constructed as a quotient of part of the original structure via an equivalence relation. An important example is a quotient group of a group. One might say
Model_theory
Mathematical group
S the relation ≤ defined by a ≤ b if and only if a'a = a'b and aa' = ba' for some a' in V(a) is a partial order in S. This is a natural partial order
Nambooripad_order
Assignment of meaning to the symbols of a formal language
first-order interpretation in which equality is interpreted by an equivalence relation and satisfies the substitution axioms for equality can be cut down
Interpretation_(logic)
On linear-time algorithms for graph logic
either they both model F or they both do not model F. This is an equivalence relation, and it can be shown by induction on the length of F that (when the
Courcelle's_theorem
Mathematical result on order relations
antisymmetric relation. A total order is a partial order that is connex. A relation R {\displaystyle R} is contained in another relation S {\displaystyle
Szpilrajn_extension_theorem
Proposed theories of gravity
Einstein's Equivalence Principle. Will tempers that by explaining experimental criteria for testing non-metric theories against Einstein's Equivalence Principle
Alternatives to general relativity
Alternatives_to_general_relativity
Mathematical operation
relations, the composition of relations is the forming of a new binary relation R ; S {\displaystyle R\mathbin {;} S} from two given binary relations R
Composition_of_relations
Mathematical logic concept
the set S contains exactly the non-negative numbers in its range. The equivalence of semidecidability and enumerability can be obtained by the technique
Computably_enumerable_set
Reasoning about equations with free variables
total, univalent relation. The facility of complementary relations inspired Augustus De Morgan and Ernst Schröder to introduce equivalences using R ¯ {\displaystyle
Algebraic_logic
Function that preserves distinctness
into Y {\displaystyle Y} . More generally, injective partial functions are called partial bijections. If f {\displaystyle f} and g {\displaystyle g}
Injective_function
Set whose pairs have minima and maxima
direction. One can now check that the relation ≤ {\displaystyle \leq } introduced in this way defines a partial ordering within which binary meets and
Lattice_(order)
Subset of incomparable elements
can define the height of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the
Antichain
Mathematical concept
an equivalence relation. Given any equivalence relation on a finite set X the associated topology is the partition topology on X. The equivalence classes
Finite_topological_space
Manifold upon which it is possible to perform calculus
atlas is taken for the range. Note that this equivalence relation is a refinement of the equivalence relation which defines a smooth manifold structure,
Differentiable_manifold
One of several equivalent definitions of a computable function
computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers
General_recursive_function
Overview of and topical guide to discrete mathematics
redirect targets Equivalence and identity Equivalence relation – Mathematical concept for comparing objects Equivalence class – Mathematical concept Equality
Outline of discrete mathematics
Outline_of_discrete_mathematics
Property that assigns truth values to k-tuples of individuals
relation is called the arity, adicity or degree of the relation. A relation with n "places" is variously called an n-ary relation, an n-adic relation
Finitary_relation
Smallest transitive relation containing a given binary relation
example, when taking the union of two equivalence relations or two preorders. To obtain a new equivalence relation or preorder one must take the transitive
Transitive_closure
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION
PARTIAL EQUIVALENCE-RELATION