For any two numbers x and y one can determine if x≤y or not. Explain why \(S\) is not an equivalence relation on \(A\). The set of rational numbers is . The relation \(R\) is symmetric and transitive. Every element of \(A\) is in its own equivalence class. It turns out that equivalence relations and partitions go hand in hand. Since \(\sim\) is an equivalence relation on \(A\), it is reflexive on \(A\). Consequently, \(\mathcal{C}\), the collection of all equivalence classes determined by \(\sim\), satisfies the first two conditions of the definition of a partition. Equivalence Relation Examples. Let \(A\) be a nonempty set and let \(\sim\) be an equivalence relation on the set \(A\). If is the equivalence relation on given by if , then is the set of circles centered at the origin. This will be explored in Exercise (12). For this equivalence relation. \(c\ S\ d\) \(d\ S\ c\). its class). Given an equivalence relation on , the set of all equivalence classes is called the {\em quotient of by }. For example, in Preview Activity \(\PageIndex{2}\), we used the equivalence relation of congruence modulo 3 on \(\mathbb{Z}\) to construct the following three sets: \[\begin{array} {rcl} {C[0]} &= & {\{a \in \mathbb{Z}\ |\ a \equiv 0\text{ (mod 3)}\},} \\ {C[1]} &= & {\{a \in \mathbb{Z}\ |\ a \equiv 1\text{ (mod 3)}\},\text{ and}} \\ {C[2]} &= & {\{a \in \mathbb{Z}\ |\ a \equiv 2\text{ (mod 3)}\}.} Consequently, each real number has an equivalence class. are the 2 distinct equivalnce classes. For example, if S is a set of numbers one relation is ≤. John Lennon and Paul McCartney, I Am the Walrus. That is, we need to show that any two equivalence classes are either equal or are disjoint. We will first prove that if \(a \sim b\), then \([a] = [b]\). The following example will show how different this can be for a relation that is not an equivalence relation. Hence 1 and 3 must be in different equivalence classes. Which of the sets \(R[a]\), \(R[b]\), \(R[c]\), \(R[d]\) and \(R[e]\) are equal? We now assume that \(y \in [b]\). Give an example of an equivalence relation R on the set A = { v, w, x, y, z } such that there are exactly three distinct equivalence classes. We can also define subsets of the integers based on congruence modulo \(n\). What are the equivalence classes for your example? Congruence modulo \(n\) is an equivalence relation on \(\mathbb{Z}\). E.g. Let \(S\) be a set. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:tsundstrom2", "Equivalence Classes", "Congruence Classes" ], https://math.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FMathematical_Logic_and_Proof%2FBook%253A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)%2F7%253A_Equivalence_Relations%2F7.3%253A_Equivalence_Classes, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), ScholarWorks @Grand Valley State University, Congruence Modulo \(n\) and Congruence Classes, \(C[0]\) consisting of all integers with a remainder of 0 when divided by 3, \(C[1]\) consisting of all integers with a remainder of 1 when divided by 3, \(C[2]\) consisting of all integers with a remainder of 2 when divided by 3. 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