Essential Information & explanations, latest texts & monographs on Equivalence.


Dynamic Equivalence: The Living Language of Christian Worship by Keith P. Pecklers

The Method of Equivalence and Its Applications (Cbms-Nsf Regional Conference Series, No 58) by Robert B. Gardner

Equivalence and Priority: Newton Versus Leibniz ; Including Leibniz's Unpublished Manuscripts on the Principia (Oxford Science Publications) by Domenico Bertoloni Meli

Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins by Francois Cardarelli

Testing Statistical Hypotheses of Equivalence by Stefan Wellek

Equivalence, Invariants and Symmetry by Peter J. Olver

Bhagavad-Gita As It Is: With the Original Sanskrit Text Roman Transliteration English Equivalence Translation and Elaborate Purports by A. C. Bhaktivedanta Swami Prabhupada

Equivalence Relations and Behavior: A Research Story by Murray Sidman

Equivalence Checking of Digital Circuits: Fundamentals, Principles, Methods by Paul Molitor

Equivalence by Shin Yu Pai

Equivalence in Measurement (Research in Management, V. 1) by Chester A. Schriesheim

The Computational Complexity of Equivalence and Isomorphism Problems (Lecture Notes in Computer Science, 1852) by Thomas Thierauf

Equivalence in Measurement (Research in Management, V. 1) by Chester Schriesheim

The Equivalence of Some Combinatorial Matching Theorems by Philip F. Reichmeider

Spaces of Homotopy Self-Equivalences: A Survey (Lecture Notes in Mathematics (Springer-Verlag), 1662) by John W. Rutter


Equivalence relation

(Redirected from Equivalence) In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.e., if the relation is written as ~ it holds for all a, b and c in X that
  • (Reflexivity) a ~ a
  • (Symmetry) if a ~ b then b ~ a
  • (Transitivity) if a ~ b and b ~ c then a ~ c
    • Equivalence relations are often used to group together objects that are similar in some sense. Table of contents showTocToggle("show","hide") 1 Examples of equivalence relations 2 Examples of relations that are not equivalences 3 Partitioning into equivalence classes 4 Generating equivalence relations 1 Common Notions in Euclid's Elements Examples of equivalence relations Examples of relations that are not equivalences
    • The relation "≥" between real numbers is not an equivalence relation, because although it is reflexive and transitive, it is not symmetric. E.g. 7 ≥ 5 does not imply that 5 ≥ 7!
    • 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).
    • The empty relation R on a non-empty set X (i.e. a R b is never true) is not an equivalence relation, because although it is vacuously symmetric and transitive, it is not reflexive (except when X is also empty).
    • 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 is not transitive (it may seem so at first sight, but many small changes can add up to a big change).
    • The relation "had a sexual intercourse" on the set of all human beings is not and equivalence relation, because although it is reflexive and symmetric, it is not transitive (if A acquired carnal knowledge of B, and B acquired carnal knowledge of C, it does not necessarily mean A acquired carnal knowledge of C)
    Partitioning into equivalence classes 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. Conversely, if the set X can be partitioned into subsets, then we can define an equivalence relation ~ on X by the rule "a ~ b if and only if a and b lie in the same subset". For example, if G is a group and H is a subgroup of G, then we can define an equivalence relation ~ on G by writing a ~ b if and only if ab-1 lies in H. The equivalence classes of this relation are the right cosets of H in G. If an equivalence relation ~ on X is given, then the set of all its equivalence classes is the quotient set of X by ~ and is denoted by X/~. Generating equivalence relations If two equivalence relations over the set X are given, then their intersection (viewed as subsets of X×X) is also an equivalence relation. This allows for a convenient way of defining equivalence relations: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, the equivalence relation ~ generated by R can be described as follows: a ~ b if and only if there exist elements x1, x2,...,xn in X such that x1 = a, xn = b and such that (xi,xi+1) or (xi+1,xi) is in R for every i = 1,...,n-1. Note that the resulting equivalence relation can often be trivial! For instance, the equivalence relation ~ generated by the binary relation <= has exactly one equivalence class: x~y for all x and y. More generally, the equivalence relation will always be trivial when generated on a relation R having the "antisymmetric" property that, given any x and y, either x R y or y R x must be true. In topology, if X is a topological space and ~ is an equivalence relation on X, then we can turn the quotient set X/~ into a topological space in a natural manner. See quotient space for the details. One often generates equivalence relations to quickly construct new spaces by "gluing things together". Consider for instance the square X = [0,1]x[0,1] and the equivalence relation on X generated by the requirements (a,0) ~ (a,1) for all a in [0,1] and (0,b) ~ (1,b) for all b in [0,1]. Then the quotient space X/~ can be naturally identified with a torus: take a square piece of paper, bend it to glue together the upper and lower edge, then bend the resulting cylinder to glue together the two mouths. Common Notions in Euclid's Elements The first person who introduced the idea of equivalence relations is Euclid in his book the Elements under Common Notions. Common Notion 1. Things which equal the same thing also equal one another. Nowadays, a binary relation is called Euclidean if it satisfies this property. Unfortunately, he didn't mention symmetry or reflexitivity. But this suggests an alternative formulation: An equivalence relation is a relation which is Euclidean, symmetric and reflexive.
    In music see octave equivalency, transpositional equivalency, inversional equivalency, enharmonic equivalency.
    The above article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License

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    Note again ... some material here is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License

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