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text: '[summary: An **equivalence relation** is a binary [-3nt] that is [reflexive_relation reflexive], [symmetric_relation symmetric], and [573 transitive]. You can think of it as a way to say when two elements of a [-3jz] are "the same" or "equivalent," despite not being literally the same element. It can be used to [set_partition partition] a set into equivalence classes.]\n\nAn **equivalence relation** is a binary [-3nt] $\\sim$ on a [-3jz] $S$ that can be used to say whether [element_of_a_set elements] of $S$ are equivalent.\n\nAn equivalence relation satisfies the following properties:\n\n1. For any $x \\in S$, $x \\sim x$ (the [reflexive_relation reflexive] property).\n2. For any $x,y \\in S$, if $x \\sim y$ then $y \\sim x$ (the [symmetric_relation symmetric] property).\n3. For any $x,y,z \\in S$, if $x \\sim y$ and $y \\sim z$ then $x \\sim z$ (the [573 transitive] property).\n\nIntuitively, any element is equivalent to itself, equivalence isn't directional, and if two elements are equivalent to the same third element then they're equivalent.\n\n## Equivalence classes\n\nWhenever we have a set $S$ with an equivalence relation $\\sim$, we can divide $S$ into *equivalence classes*, i.e. sets of mutually equivalent elements. The equivalence class of some $x \\in S$ is the set of elements of $S$ equivalent to $x$, written $[x]=\\{y \\in S \\mid x \\sim y\\}$, and we say that $x$ is a "representative" of $[x]$. We call the set of equivalence classes $S/\\sim = \\{[x] \\mid x \\in S\\}$.\n\nFrom the definition of an equivalence relation, it's easy to show that $x \\in [x]$ and $[x]=[y]$ if and only if $x \\sim y$.\n\nIf you have a set already [set_partition partitioned] into subsets ($S$ is the [-disjoint_union] of the elements of a collection $A$), you can go the other way and define a relation with two elements related whenever they're in the same subset ($x \\sim y$ when there is some $U \\in A$ with $x,y \\in U$). Then this is an equivalence relation, and the partition of the set is the set of equivalence classes under this relation (that is, $[x] \\in A$ and $A=S/\\sim$).\n\n##Defining functions on equivalence classes\n\nSuppose you have a [-3jy] $f: S \\to U$ and want to define a corresponding function $f^*: S/\\sim \\to U$, where $U$ can be any set. You define $f^*([x])$ to be $f(x)$. This could be a problem; what if $x \\sim y$ but $f(x) \\neq f(y)$? Then $f^*([x])=f^*([y])$ wouldn't be [-well_defined well-defined]. Whenever you define a function on equivalence classes in terms of representatives, you have to make sure the definition doesn't depend on which representative you happen to pick. In fact, one way you might arrive at an equivalence relation is to say that $x \\sim y$ whenever $f(x)=f(y)$.\n\nIf you have a function $f: S \\to S$ and want to define $f^*: S/\\sim \\to S/\\sim$ by $f^*([x])=[f(x)]$, you have to verify that whenever $x \\sim y$, $[f(x)]=[f(y)]$, equivalently $f(x) \\sim f(y)$. \n\n#Examples\n\nConsider the [48l integers] with the relation $x \\sim y$ if $n|x-y$, for some $n \\in \\mathbb N$. This is the integers [modular_arithmetic mod] $n$. The [-addition] and [-multiplication] operations can be inherited from the integers, so it makes sense to talk about addition and multiplication mod $n$. \n\nThe [4bc real numbers] can be defined as equivalence classes of [50d Cauchy sequences].\n\n[4f4] is an equivalence relation (unless the objects form a [-proper_class]).',
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