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In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of elements that are related to one another, forming what are called equivalence classes. Notationally, given a set and an equivalence relation on , the ''equivalence class'' of an element in is the subset of all elements in which are equivalent to . It follows from the definition of the equivalence relations that the equivalence classes form a . The set of equivalence classes is sometimes called the quotient set or the quotient space of by and is denoted by . When has some structure, and the equivalence relation is defined with some connection to this structure, the quotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. ==Notation and formal definition== An equivalence relation is a binary relation satisfying three properties: *For every element in , (reflexivity), *For every two elements and in , if , then (symmetry) *For every three elements , , and in , if and , then (transitivity). The equivalence class of an element is denoted and is defined as the set : of elements that are related to by . An alternative notation can be used to denote the equivalence class of the element , specifically with respect to the equivalence relation . This is said to be the -equivalence class of . The set of all equivalence classes in with respect to an equivalence relation is denoted as and called modulo (or the quotient set of by ). The surjective map from onto , which maps each element to its equivalence class, is called the canonical surjection or the canonical projection map. When an element is chosen (often implicitly) in each equivalence class, this defines an injective map called a ''section''. If this section is denoted by , one has for every equivalence class . The element is called a representative of . Any element of a class may be chosen as a representative of the class, by choosing the section appropriately. Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called ''canonical representatives''. For example, in modular arithmetic, consider the equivalence relation on the integers defined by if is a multiple of a given integer , called the ''modulus''. Each class contains a unique non-negative integer smaller than , and these integers are the canonical representatives. The class and its representative are more or less identified, as is witnessed by the fact that the notation may denote either the class or its canonical representative (which is the remainder of the division of by ). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Equivalence class」の詳細全文を読む スポンサード リンク
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