Equivalence class
Overview
- This article is about equivalency in mathematicsMathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
; for equivalency in musicMusicMusic is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
see equivalence class (music)Equivalence class (music)In music theory, equivalence class is an equality or equivalence between sets or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal...
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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, given a set X and an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
~ on X, the equivalence class of an element a in X is the subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of all elements in X which are equivalent to a:
The notion of equivalence classes is useful for constructing sets out of already constructed ones.
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