Search results
Results from the WOW.Com Content Network
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).
An equivalence relation on a set is a binary relation on satisfying the three properties: [1] for all (reflexivity), implies for all (symmetry), if and then for all (transitivity). The equivalence class of an element is defined as [2] The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although ...
Hyperfinite equivalence relation. In descriptive set theory and related areas of 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 sense, be approximated by Borel equivalence relations that have finite classes.
Congruence relation. In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. [1]
The minimal automaton accepting our language would have three states corresponding to these three equivalence classes. Another immediate corollary of the theorem is that if for a language the relation has infinitely many equivalence classes, it is not regular. It is this corollary that is frequently used to prove that a language is not regular.
Graph isomorphism. In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H. such that any two vertices u and v of G are adjacent in G if and only if and are adjacent in H. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism ...
The congruence relation is an equivalence relation. The equivalence class modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted as (a mod m), or as a or [a] when the modulus m is known from the context.
Then the relation x R y is equivalent with the equality x R = y R. It follows that equality is the finest equivalence relation on any set S in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element). In some contexts, equality is sharply distinguished from equivalence or ...