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# Topic: Equivalence relation

###### In the News (Fri 19 Jul 19)

 Equivalence Relation Equivalence relation, a mathematical concept, is a type of relation on a given set that provides a way for elements of that set to be identified with (meaning considered equivalent to for some present purpose) other elements of that set. Equivalence relation is defined in a branch of mathematics called set theory, a vital branch underpinning all branches of mathematics and those fields that use mathematics. Equivalence relations are so ubiquitous in mathematics and other fields that use mathematics because they enable the user to partition a set in a particular way of the user’s design. www.iscid.org /encyclopedia/Equivalence_Relation   (2634 words)

 Equivalence relation - Wikipedia, the free encyclopedia The equivalence kernel of a bijection is the identity relation. Green's relations are five equivalence relations on the elements of a semigroup. Hence an equivalence relation is a relation that is Euclidean and reflexive. en.wikipedia.org /wiki/Equivalence_relation   (2425 words)

 PlanetMath: equivalence relation An equivalence relation on a set induces a partition on it, and also any partition induces an equivalence relation. This is version 8 of equivalence relation, born on 2001-10-18, modified 2004-02-14. For example, the integers modulo n are partitioned into n equivalence classes, by the relation a R b iff remainder(a/n)=remainder(b/n). planetmath.org /encyclopedia/EquivalenceRelation.html   (143 words)

 NationMaster - Encyclopedia: Equivalence relation   (Site not responding. Last check: ) In mathematics, an equivalence relation, denoted by an infix "~", is a binary relation on a set X that is reflexive, symmetric, and transitive. The equivalence kernel of a bijection is the identity relation. Hence an equivalence relation is a relation that is Euclidean and reflexive. www.nationmaster.com /encyclopedia/Equivalence_relation   (0 words)

 NationMaster - Encyclopedia: Quotient set   (Site not responding. Last check: ) In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: This equivalence relation is known as the kernel of f. Because of the properties of an equivalence relation it holds that a is in [a] and that any two equivalence classes are either equal or disjoint. www.nationmaster.com /encyclopedia/Quotient-set   (0 words)

 Equivalence relation   (Site not responding. Last check: ) Equivalence relations are often used to group together objects that are similar in some sense. The relation "has a common factor with" between natural numbers is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (2 and 6 have a common factor, and 6 and 3 have a common factor, but 2 and 3 do not have a common factor). Every equivalence relation on X defines a partition of X into subsets called equivalence classes: all elements equivalent to each other are put into one class. www.ebroadcast.com.au /lookup/encyclopedia/eq/Equivalent.html   (738 words)

 Equivalence class - Wikipedia, the free encyclopedia This equivalence relation is known as the kernel of f. Because of the properties of an equivalence relation it holds that a is in [a] and that any two equivalence classes are either equal or disjoint. If ~ is an equivalence relation on X, and P(x) is a property of elements of x, such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~. en.wikipedia.org /wiki/Equivalence_class   (866 words)

 PlanetMath: real number This definition is well-defined and does not depend on the choice of Cauchy sequences used to represent the equivalence classes. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield. planetmath.org /encyclopedia/MathbbR.html   (919 words)

 Math Forum - Ask Dr. Math Define a relation R on the power set of X by A R B if A U Y = B U Y. a) Prove that R is an equivalence relation. Recall that equivalence classes are distinct (their intersections are always empty) so you'll never use any of the four that we found above in any other class. Certainly two distinct elements of P(X) won't be in the same equivalence class under this new definition of the relation ~ because this can only happen when the elements are related (i.e., they are equal) and we are assuming from the outset that these two elements are distinct. www.mathforum.org /library/drmath/view/51864.html   (0 words)

 Equivalence Relation   (Site not responding. Last check: ) Equivalence relations can also be represented by a digraph since they are a binary relation on a set. The set of even numbers and that of odd numbers in the equivalence relation of congruent mod 2, and the set of integers equivalent to a number between 1 and 12 in the equivalence relation on hours in the clock example are called an equivalence class. Definition(equivalence class): For an equivalence relation R on a set A, the set of the elements of A that are related to an element, say a, of A is called the www.cs.odu.edu /~toida/nerzic/content/relation/eq_relation/eq_relation.html   (649 words)

 EQUIVALENCE RELATION - GoGoSearch.com In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric, and transitive. Equivalence relations are often used to group together objects that are similar in some sense. Green's relations are five equivalence relations on the elements of a semigroup. www.gogosearch.com /wiki/equivalence_relation   (886 words)

 Equivalence relation Summary In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric, and transitive. Green's relations are five equivalence relations on the elements of a semigroup. The relation "is a sibling of" on the set of all human beings is not an equivalence relation, and it is worthwhile example to consider. www.bookrags.com /Equivalence_relation   (2223 words)

 Equivalence Relation   (Site not responding. Last check: ) Equivalence relations can also be represented by a digraph since they are a binary relation on a set. The set of even numbers and that of odd numbers in the equivalence relation of congruent mod 2, and the set of integers equivalent to a number between 1 and 12 in the equivalence relation on hours in the clock example are called an equivalence class. Definition(equivalence class): For an equivalence relation R on a set A, the set of the elements of A that are related to an element, say a, of A is called the cs.odu.edu /~toida/nerzic/content/relation/eq_relation/eq_relation.html   (649 words)

 Relations - PineWiki An equivalence relation is a relation that is reflexive, symmetric, and transitive. Any equivalence relation ~ on a set A gives rise to a set of equivalence classes, where the equivalence class of an element a is the set of all b such that a ~ b. The reflexive closure of a relation R (whose domain and codomain are equal) is the smallest super-relation of R that is reflexive; it is obtained by adding (x,x) to R for all x in R's domain. pine.cs.yale.edu /pinewiki/Relations   (2456 words)

 Equivalence relation : Equivalent   (Site not responding. Last check: ) In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. The relation "has a common factor with" between natural numbers is not an equivalence relation, because although it is reflexive and symmetric, it isn't transitive (2 and 6 have a common factor, and 6 and 3 have a common factor, but 2 and 3 do not have a common factor). The relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not an equivalence relation, because although it is reflexive and symmetric, it isn't transitive (it may seem so at first sight, but many small changes can add up to a big change). www.explainthat.info /eq/equivalent.html   (962 words)

 Relations : Software Foundations : Thomas Alspaugh : UCI is a relation that is reflexive, symmetric, and transitive. These equivalence classes are disjoint (their intersection is empty) and exhaustive (every element is in an equivalence class). ) is a relation that is antisymmetric and transitive. www.ics.uci.edu /~alspaugh/foundations/relation.html   (1159 words)

 Equivalence Relation A relation possessing all three properties is called an equivalence relation. The relation r is a partial ordering on the set s, or s is a partially ordered set via r, or s is a poset, if r is transitive and antisymmetric. Equivalently, a complete lattice is a complete relation with a bound on every set. www.mathreference.com /set,rst.html   (1110 words)

 Equivalence Relationship Equivalence relation is a binary (in the sense that it relates two elements) relation that satisfies the following three properties: If you are indeed curious, try to think of other properties common to all equivalence relations you may think of. be an equivalence relation defined between elements of a set A. www.cut-the-knot.org /blue/equi.shtml   (760 words)

 Reference.com/Encyclopedia/Equivalence class Consider the "modulo 2" equivalence relation on the set Z of integers: x~y if and only if x-y is even. The equivalence classes are known as right cosets of H in G; one of them is H itself. If ~ is an equivalence relation on X, and P(x) is a property of elements of x, such that whenever x ~ y, P(x) is true if P(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~. www.reference.com /browse/wiki/Quotient_set   (0 words)

 Equivalence Relations and Equivalence Classes   (Site not responding. Last check: ) A relation R tells for any two members, say x and y, of S whether x is in that relation to y. The power of the concept of equivalence class is that operations can be defined on the equivalence classes using representatives from each equivalence class. For fractions, (a/b) is equivalent to (c/d) if one can be represented in the form in which its components are a constant multiple of the components of the other, say (c/d)=(ka/kb). www2.sjsu.edu /faculty/watkins/equivalence.htm   (285 words)

 ABSTRACT ALGEBRA: OnLine Study Guide, Section 2.2 Let ~ be an equivalence relation on the set S. For a given element a in S, we define the equivalence class of a to be the set of all elements of S that are equivalent to a. Equivalence relations are important because in a wide variety of situations it is useful to split a set up into subsets in which the elements have some property in common. First there is the definition of an equivalence relation on S, which tells you when two different elements of S have the same properties and therefore belong to the same subset. www.math.niu.edu /~beachy/abstract_algebra/study_guide/22.html   (994 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: ) Define a relation R on the power set of X by A R B if A U Y = B U Y. a) Prove that R is an equivalence relation. Recall that equivalence classes are distinct (their intersections are always empty) so you'll never use any of the four that we found above in any other class. Certainly two distinct elements of P(X) won't be in the same equivalence class under this new definition of the relation ~ because this can only happen when the elements are related (i.e., they are equal) and we are assuming from the outset that these two elements are distinct. mathforum.org /library/drmath/view/51864.html   (698 words)

 PlanetMath: integer is defined to be the set of equivalence classes of pairs of natural numbers The ring of integers is also a Euclidean domain, with valuation given by the absolute value function. Cross-references: function, absolute value, valuation, Euclidean domain, ring of integers, ordered ring, ordering relation, integral domain, operations, representation, class, multiplication, addition, equivalence relation, equivalence classes, negatives, natural numbers planetmath.org /encyclopedia/Integer.html   (182 words)

 Equivalence relations In geometry, similarity of triangles is an equivalence relation. Thus the equivalence classes of a relation are a partition. Each equivalence relation on a set partitions the set into its equivalence classes but also for each partition of the set there is an equivalence relation whose equivalence classes are the sets in the partition. www.math.csusb.edu /notes/rel/node3.html   (0 words)

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