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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 ...
The row and column indices of nonwhite cells are the related elements, while the different colors, other than light gray, indicate the equivalence classes (each light gray cell is its own equivalence class). In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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]
Conjugacy class. Two Cayley graphs of dihedral groups with conjugacy classes distinguished by color. In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes.
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.
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 all axioms of the real numbers. Q {\displaystyle \mathbb {Q} } can be considered as a subset of R {\displaystyle \mathbb {R} } by identifying a rational number r with the equivalence class ...
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 ...
The quotient space is then defined as , the set of all equivalence classes induced by on . Scalar multiplication and addition are defined on the equivalence classes by [2][3] for all , and. . It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives). These operations turn the quotient ...