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Mathematical concept
notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle S} into equivalence classes. These equivalence
Equivalence_class
Mathematical concept for comparing objects
{\displaystyle a=c} (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given
Equivalence_relation
Topics referred to by the same term
(mathematics) Equivalence relation Equivalence class Equivalence of categories, in category theory Equivalent infinitesimal Identity Matrix equivalence in linear
Equivalence
Software testing technique
Equivalence partitioning or equivalence class partitioning (ECP) is a software testing technique that divides the input data of a software unit into partitions
Equivalence_partitioning
Enharmonic equivalence In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows
Equivalence_class_(music)
A forwarding equivalence class (FEC) is a term used in Multiprotocol Label Switching (MPLS) to describe a set of packets with similar or identical characteristics
Forwarding_equivalence_class
In group theory, equivalence class under the relation of conjugation
b=gag^{-1}.} This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under the
Conjugacy_class
Number in {..., –2, –1, 0, 1, 2, ...}
independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form (n,0) or (0,n)
Integer
Generalization of "n-th" to infinite cases
bijections, being order-isomorphic is an equivalence relation on well-ordered sets; its equivalence classes correspond to order types (ordinals). In the
Ordinal_number
Mathematical equivalence relation
matrix equivalence a generalization of row equivalence. The rank property yields an intuitive canonical form for matrices of the equivalence class of rank
Matrix_equivalence
Generalization of the real numbers
an equivalence class of representations of the form { L | R } that designate the same number, noting that each equivalence class is a proper class rather
Surreal_number
Approximation of a mathematical set
{\displaystyle X} comprises a single class, and we wish to express this class (i.e., this subset) using the equivalence classes induced by attribute subset P
Rough_set
Quadratic homogeneous polynomial in two variables
partitioned into equivalence classes, called classes of quadratic forms. A class invariant can mean either a function defined on equivalence classes of forms
Binary_quadratic_form
Topological space construction
be an equivalence relation on X . {\displaystyle X.} The quotient set Y = X / ∼ {\displaystyle Y=X/{\sim }} is the set of equivalence classes of elements
Quotient_space_(topology)
Continuous deformation between two continuous functions
states. Fiber-homotopy equivalence (relative version of a homotopy equivalence) Homeotopy Homotopy type theory Mapping class group Poincaré conjecture
Homotopy
Topological relational characteristic
write x ≡ y for "x and y are topologically indistinguishable". The equivalence class of x will be denoted by [x]. By definition, any two distinct points
Topological indistinguishability
Topological_indistinguishability
Equivalence class in mathematics
In combinatorics, a k-ary necklace of length n is an equivalence class of n-character strings over an alphabet of size k, taking all rotations as equivalent
Necklace_(combinatorics)
Quotient of two integers
equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (m, n) in the equivalence class
Rational_number
Aspect of the Unicode standard
Unicode equivalence is the specification by the Unicode character encoding standard that some sequences of code points represent essentially the same
Unicode_equivalence
Vector space consisting of affine subsets
vectors in U {\displaystyle U} get mapped into the equivalence class of the zero vector. The equivalence class – or, in this case, the coset – of v {\displaystyle
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Reflexive and transitive binary relation
together with a partial order on the set of equivalence class, cf. picture. Like partial orders and equivalence relations, preorders (on a nonempty set)
Preorder
Mathematical statement of uniqueness, except for an equivalent structure
if they have the same equivalence class, it is also the case that a and b are equal up to R if and only if the equivalence classes of a and b with respect
Up_to
Generalizations of codimension-1 subvarieties of algebraic varieties
1-forms yield linearly equivalent divisors. Any divisor in this linear equivalence class is called the canonical divisor of X, KX. The genus g of X can be
Divisor_(algebraic_geometry)
Informal use of mathematical notation
"there are two isomorphism classes of non-Abelian groups of order 8". Referring to an equivalence class of an equivalence relation by x {\displaystyle
Abuse_of_notation
In number theory, measure of non-unique factorization
shown that this is an equivalence relation. The equivalence classes are called the ideal classes of R {\displaystyle R} . Ideal classes can be multiplied:
Ideal_class_group
Movement with a fixed point is rotation
is a class function. This matrix transformation is clearly an equivalence relation, that is, all such equivalent matrices form an equivalence class. In
Euler's_rotation_theorem
Mathematical construction
M_{\bullet }.} Explicitly, if the U {\displaystyle {\mathcal {U}}} -equivalence class of an element a ∈ ∏ i ∈ I M i {\displaystyle a\in {\textstyle \prod
Ultraproduct
Equivalence class of objects sharing local properties at a point in a topological space
the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared
Germ_(mathematics)
Number representing a continuous quantity
ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite
Real_number
Number system extending the rational numbers
numbers may also be defined as equivalence classes, in a similar way as the definition of real numbers as equivalence classes of Cauchy sequences. It is fundamentally
P-adic_number
Branch of music theory
If x is a pitch class, the inversion with index number n is written In = n - x mod 12. "For a relation in set S to be an equivalence relation [in algebra]
Set_theory_(music)
Study of angle-preserving transformations of a geometric space
called the conformal factor. An equivalence class of such metrics is known as a conformal metric or conformal class. Thus, a conformal metric may be
Conformal_geometry
Arithmetic operation
the equivalence class of the result depends only on the equivalence classes of the summands, and thus that this defines an addition of equivalence classes
Addition
Metric geometry
{\displaystyle M'} with the equivalence class of sequences in M {\displaystyle M} converging to x {\displaystyle x} (i.e., the equivalence class containing the sequence
Complete_metric_space
Indication of topological symmetry groups to topological condensed matter
given a symmetry class and a dimension of the Brillouin zone, what are all the equivalence classes of Hamiltonians. Each equivalence class can be labeled
Periodic table of topological insulators and topological superconductors
Periodic_table_of_topological_insulators_and_topological_superconductors
prefix equivalence class if the string representation of them are identical up to the (k-1)-th node. In other words, all elements in a prefix equivalence class
Frequent_subtree_mining
Equivalence relation in algebra
corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. The definition of a congruence depends
Congruence_relation
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
Object within another object of the same category
v\leq u} is an equivalence relation on the monomorphisms with codomain A {\displaystyle A} , and the corresponding equivalence classes of these monomorphisms
Subobject
Necessary and sufficient condition for a formal language to be regular
{\displaystyle \sim _{L}} is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes. The Myhill–Nerode theorem
Myhill–Nerode_theorem
Set theory method
trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the
Scott's_trick
Geometric object that has length and direction
form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector. The equivalence class of (A, B) is often denoted
Euclidean_vector
Kind of partial function between algebraic varieties
V → W {\displaystyle f\colon V\to W} between two varieties is an equivalence class of pairs ( f U , U ) {\displaystyle (f_{U},U)} in which f U {\displaystyle
Rational_mapping
Mathematical notion of infinitesimal difference
equivalence class [f − f(p)] in the quotient space Ip/Ip2, and the 1-jet of f (which encodes its value and its first derivative) is the equivalence class
Differential_(mathematics)
Property that is not changed by mathematical transformations
generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas
Invariant_(mathematics)
Projective line over the real numbers
all equivalence classes. Each equivalence class is considered as a single point, or, in other words, a point is defined as being an equivalence class. If
Real_projective_line
Method to measure temperature quantitatively
in form of an equivalence relation. Accordingly, the set of all thermal systems may be divided into a quotient set of equivalence classes, denoted as M
Scale_of_temperature
Square array with symbols that each occur once per row and column
fact, equal. Isomorphism is also an equivalence relation and its equivalence classes are called isomorphism classes. Another type of operation is easiest
Latin_square
Abelian group related to division algebras
group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor
Brauer_group
Operation in graph theory
local equivalence class. The local equivalence classes on graphs with up to 12 vertices has been computed. The size of a local equivalence class is at
Local_complementation
2x2x2 combination puzzle
its equivalence class. The quotient set A M / ∼ {\displaystyle A_{M}/\sim } can be formed using these equivalence classes. It contains the equivalence classes
Pocket_Cube
Maximal subgraph whose vertices can reach each other
belongs to exactly one equivalence class. The components are then the induced subgraphs formed by each of these equivalence classes. Alternatively, some
Component_(graph_theory)
nested parentheses. 2. Equivalence class: given an equivalence relation, [ x ] {\displaystyle [x]} often denotes the equivalence class of the element x. 3
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Mathematical ranking of a set
object (specifically, they are identified together in their common equivalence class). Definition A strict weak ordering on a set S {\displaystyle S} is
Weak_ordering
Manifold upon which it is possible to perform calculus
A tangent vector at p ∈ M is an equivalence class of differentiable curves γ with γ(0) = p, modulo the equivalence relation of first-order contact between
Differentiable_manifold
Statistical shape analysis technique
points. The shape of an object can be considered as a member of an equivalence class formed by removing the translational, rotational and uniform scaling
Procrustes_analysis
Coordinate system used in projective geometry
∼ {\displaystyle \sim } is an equivalence relation and the projective plane can be defined as the equivalence classes of R 3 ∖ { 0 } . {\displaystyle
Homogeneous_coordinates
Mapping which preserves all topological properties of a given space
Being "homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes. The third requirement, that
Homeomorphism
Topological space that is connected
the whole space. In fact, a connected component is the same as an equivalence class when two points are equivalent if they belong to the same connected
Connected_space
Category where every morphism is invertible; generalization of a group
with the equivalence relation ∼ {\displaystyle \sim } is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an
Groupoid
Euclidean space without distance and angles
parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction. Unlike for vectors
Affine_space
Topics referred to by the same term
the size of the ideal class group of a number ring Class number (binary quadratic forms), the number of equivalence classes of binary quadratic forms
Class_number
Mathematical ways to group elements of a set
exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped
Partition_of_a_set
This means that x, y and z are equivalent – they are in the same equivalence class of the quotient. In other words, H 0 ( X ) {\displaystyle H_{0}(X)}
Graph_homology
In mathematics, invertible homomorphism
is an equivalence relation. An equivalence class given by isomorphisms is commonly called an isomorphism class. Examples of isomorphism classes are plentiful
Isomorphism
Algebraic construction
π : A → A / B {\displaystyle \pi :A\to A/B} sending a in A to its equivalence class a + B is called the quotient map or the projection map, and is a module
Quotient_module
Completion of the usual space with "points at infinity"
this set of vector lines. This set can be the set of equivalence classes under the equivalence relation between vectors defined by "one vector is the
Projective_space
Process for anonymizing data
sensitive field in each equivalence class exist. Entropy l-diversity – The most complex definition defines Entropy of an equivalent class E to be the negation
L-diversity
Homological construction
which sends each morphism to its chain homotopy equivalence class. Since every chain homotopy equivalence is a quasi-isomorphism, Q {\displaystyle Q} factors
Derived_category
Theories in mathematical logic
first-order properties of equivalence relations are: ~ has an infinite number of equivalence classes; ~ has exactly n equivalence classes (for any fixed positive
List_of_first-order_theories
Algebraic geometry scheme
identified with the Picard group Pic(U). As a result, KU defines a linear equivalence class of Weil divisors on X. Any such divisor is called the canonical divisor
Gorenstein_scheme
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
Maximal biconnected subgraph
and only if they belong to the same equivalence class. The subgraphs formed by the edges in each equivalence class are the biconnected components of the
Biconnected_component
Cohomology with real coefficients computed using differential forms
Poincaré lemma. The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold. One classifies two closed forms α
De_Rham_cohomology
Form of an object
is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets
Shape
This defines an equivalence relation that is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy
Construction of the real numbers
Construction_of_the_real_numbers
result on Clifford modules is that the Morita equivalence class of a Clifford algebra (the equivalence class of the category of Clifford modules over it)
Clifford_module
Probability theory term
Radon–Nikodym derivative) as an equivalence class of functions (of x). The right choice of functions within these equivalence classes may be made as above; it
Conditioning_(probability)
3-volume treatise on mathematics, 1910–1913
properties of cardinals. A cardinal is defined to be an equivalence class of similar classes (as opposed to ZFC, where a cardinal is a special sort of
Principia_Mathematica
relation " ∼ {\displaystyle \sim } " is an equivalence relation in the set of all IBQFs. The equivalence class to which the IBQF g ( x , y ) {\displaystyle
Gauss_composition_law
Computing paradigm based on information entities ("granules")
classified into classes of [ x ] Q {\displaystyle [x]_{Q}} based on knowledge of [ x ] P . {\displaystyle [x]_{P}.} The equivalence classes of [ x ] Q {\displaystyle
Granular_computing
Procedure of coping with redundant degrees of freedom in physical field theories
the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by
Gauge_fixing
in the same equivalence class. There are three fundamental approaches to constructing measures of network similarity: structural equivalence, automorphic
Similarity_(network_science)
Topological space that locally resembles Euclidean space
manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single
Manifold
mathematics, a hyperfinite equivalence relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that can, in a certain
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
Function whose squared absolute value has finite integral
functions (in the sense mentioned in which a "function" actually means an equivalence class of functions that are equal almost everywhere) form an inner product
Square-integrable_function
of each finite cardinality (the equivalence classes themselves are too large to be sets); in NFU the equivalence classes themselves are sets, and are thus
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Mathematical concept
S_{c}} .) Thus, S c {\displaystyle S_{c}} partitions into equivalence classes. Each equivalence class comprises a collection of quadratic irrationalities with
Quadratic_irrational_number
Mathematical function such that every output has at least one input
followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently
Surjective_function
Group obtained by aggregating similar elements of a larger group
operates on each such class (known as a congruence class) as a single entity. For a congruence relation on a group, the equivalence class of the identity element
Quotient_group
Size of a set in mathematics
equivalent to one another. These equivalence classes are called the Frege–Russell cardinal numbers. However, these equivalence classes are too large to form sets
Cardinality
and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes. Two permutation classes are Wilf equivalent when they have
Wilf_equivalence
Graph operation
elements that are associates of each other is an equivalence class, and the element of each equivalence class that has a > 0 {\displaystyle a>0} and b ≥ 0
Goldberg–Coxeter_construction
Logic puzzle
from the appropriate equivalence class. Because the actual sequence and the representative sequence are in the same equivalence class, their entries are
Induction_puzzles
Topics referred to by the same term
wagering for a horse to win or finish second Place (mathematics), an equivalence class of absolute values of an integral domain or field In place-value,
Place
Generalization of strings in computer science
In computer science, a trace is an equivalence class of strings, wherein certain letters in the string are allowed to commute, but others are not. Traces
Trace_monoid
Task of transforming a deterministic finite automaton
terminates. Lemma. Given a fixed character c and an equivalence class Y that splits into equivalence classes B and C, only one of B or C is necessary to refine
DFA_minimization
Cohomology theory for Lie algebras
+ 1 ( g ; M ) {\displaystyle H^{n+1}({\mathfrak {g}};M)} gives an equivalence class of ways to extend the Lie algebra g {\displaystyle {\mathfrak {g}}}
Lie_algebra_cohomology
an equivalence relation on the set of geodesic rays, and the set of equivalence classes is called the ideal boundary ∂X of X. An equivalence class of
Tits_metric
Generalization of equivalence classes to scheme theory
sheaves. Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X. Then the map q : X → Q {\displaystyle
Quotient by an equivalence relation
Quotient_by_an_equivalence_relation
EQUIVALENCE CLASS
EQUIVALENCE CLASS
Girl/Female
American, British, English
Feminine Equivalent of Count; Titled
Boy/Male
Muslim American Biblical English Hebrew
The Biblical Adam is the English language equivalent.
Boy/Male
Muslim
The Biblical Elijah is the English language equivalent.
Girl/Female
Hawaiian
Sound. Also the Hawaiian equivalent of Sandy.
Girl/Female
Hawaiian American Latin Slavic Spanish
Equivalent of English Lara: famous;cheerful.
Girl/Female
English
Titled. Feminine equivalent of Count.
Boy/Male
Muslim
The Biblical Ezekiel is the English language equivalent.
Boy/Male
Muslim
The Biblical Jacob is the English language equivalent.
Boy/Male
Indian, Sanskrit
Equivalent to Ten Heavens; Very Powerful
Boy/Male
Muslim
The Biblical Adam is the English language equivalent.
Girl/Female
Indian, Sanskrit
Equivalent to a God
Boy/Male
Scandinavian
Royalty title approximately equivalent to the English Earl.
Boy/Male
Muslim
The Biblical Jethro is the English language equivalent.
Boy/Male
Muslim
The Biblical Jacob is the English language equivalent.
Boy/Male
Muslim
The Biblical Ezra is the English language equivalent.
Boy/Male
Muslim
The Biblical Elijah is the English language equivalent.
Boy/Male
Muslim
The Biblical Abel is the English language equivalent.
Boy/Male
Muslim
The Biblical Elijah is the English language equivalent.
Boy/Male
Muslim
Ezekiel (English language equivalent).
Boy/Male
Muslim
The Biblical Areran is the English language equivalent.
EQUIVALENCE CLASS
EQUIVALENCE CLASS
Girl/Female
Tamil
Unity
Boy/Male
Tamil
Wealthy
Boy/Male
American, Anglo, British, English
Brave Friend
Girl/Female
Hindu, Indian
Just Like Mother who Take Care of Others
Male
Italian
 Italian name ARMO means "crew." Compare with another form of Armo.
Girl/Female
Muslim/Islamic
Clean pure
Male
Native American
Native American Algonquin name ENKOODABOOAOO means "one who lives alone."
Boy/Male
Tamil
Agastya | அகஸà¯à®¤à¯à®¯à®¾
Name of a sage
Girl/Female
Tamil
Pallabi | பலà¯à®²à®¾à®ªà¯€ Â
Leaf
Girl/Female
Bengali, Danish, Gujarati, Hindu, Indian, Persian, Swedish
One who Brings Joy; Moving; Help; Light; Glow; Goddess Sita
EQUIVALENCE CLASS
EQUIVALENCE CLASS
EQUIVALENCE CLASS
EQUIVALENCE CLASS
EQUIVALENCE CLASS
a.
Equal in wortir or value, force, power, effect, import, and the like; alike in significance and value; of the same import or meaning.
n.
Equal weight; equiponderance.
a.
Equal; equivalent; adequate.
n.
That comparative quantity by weight of an element which possesses the same chemical value as other elements, as determined by actual experiment and reference to the same standard. Specifically: (a) The comparative proportions by which one element replaces another in any particular compound; thus, as zinc replaces hydrogen in hydrochloric acid, their equivalents are 32.5 and 1. (b) The combining proportion by weight of a substance, or the number expressing this proportion, in any particular compound; as, the equivalents of hydrogen and oxygen in water are respectively 1 and 8, and in hydric dioxide 1 and 16.
a.
Equal in measure but not admitting of superposition; -- applied to magnitudes; as, a square may be equivalent to a triangle.
prep.
Denoting identity or equivalence; -- used with a name or appellation, and equivalent to the relation of apposition; as, the continent of America; the city of Rome; the Island of Cuba.
n.
A combining unit, whether an atom, a radical, or a molecule; as, in acid salt two or more equivalents of acid unite with one or more equivalents of base.
n.
The condition of being equivalent or equal; equality of worth, value, signification, or force; as, an equivalence of definitions.
p. pr. & vb. n.
of Equibalance
imp. & p. p.
of Equibalance
n.
Equal power or force; equivalent amount.
n.
The degree of combining power as determined by relative weight. See Equivalent, n., 2.
v. t.
To be equivalent or equal to; to counterbalance.
a.
Contemporaneous in origin; as, the equivalent strata of different countries.
v. t.
To make of equal weight; to balance equally; to counterbalance; to equiponderate.
n.
Something equivalent; that which is equal in value, worth, weight, or force; as, to offer an equivalent for damage done.
a.
Equivalent in value, signification, or effect.
n.
The quantity of the combining power of an atom, expressed in hydrogen units; the number of hydrogen atoms can combine with, or be exchanged for; valency. See Valence.
n.
Same as Equivalence.
v. t.
To make the equivalent to; to equal; equivalence.