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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 ...
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]
In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. Equality between A and B is written A = B, and pronounced " A equals B ". In this equality, A and B are the members ...
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.
Logical biconditional. In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement " if and only if " (often abbreviated as " iff " [1]), where is known as the antecedent, and the ...
Logical equivalence. In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of and is sometimes expressed as , , , or , depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation ...
Equivalence relationships exist between exact copies of the same manifestation of a work or between an original item and reproductions of it, so long as the intellectual content and authorship are preserved. Examples include reproductions such as copies, issues, facsimiles and reprints, photocopies, and microfilms.