# equivalence class in relation

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The relation $$S$$ defined on the set $$\{1,2,3,4,5,6\}$$ is known to be $\displaylines{ S = \{ (1,1), (1,4), (2,2), (2,5), (2,6), (3,3), \hskip1in \cr (4,1), (4,4), (5,2), (5,5), (5,6), (6,2), (6,5), (6,6) \}. / Some authors use "compatible with ~" or just "respects ~" instead of "invariant under ~". Cem Kaner  defines equivalence class as follows: If you expect the same result 5 … We have demonstrated both conditions for a collection of sets to be a partition and we can conclude We have indicated that an equivalence relation on a set is a relation with a certain combination of properties (reflexive, symmetric, and transitive) that allow us to sort the elements of the set into certain classes. {a\mathop {R} b}} Equivalence classes are an old but still central concept in testing theory. [ , Let $$A$$ be a set with partition $$P=\{A_1,A_2,A_3,...\}$$ and $$R$$ be a relation induced by partition $$P.$$ WMST $$R$$ is an equivalence relation. Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X. X b Find the equivalence relation (as a set of ordered pairs) on $$A$$ induced by each partition. := A relation R on a set A is said to be an equivalence relation if and only if the relation R is reflexive, symmetric and transitive. Equivalently. a Since $$a R b$$, we also have $$b R a,$$ by symmetry. Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, 6.3: Equivalence Relations and Partitions, [ "article:topic", "authorname:hkwong", "license:ccbyncsa", "showtoc:yes", "equivalence relation", "Fundamental Theorem on Equivalence Relation" ], https://math.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FCourses%2FMonroe_Community_College%2FMATH_220_Discrete_Math%2F6%253A_Relations%2F6.3%253A_Equivalence_Relations_and_Partitions, $$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, \[a\sim b \,\Leftrightarrow\, a \mbox{ mod } 4 = b \mbox{ mod } 4$, $a\sim b \,\Leftrightarrow\, a \mbox{ mod } 3 = b \mbox{ mod } 3.$, $S_i \sim S_j\,\Leftrightarrow\, |S_i|=|S_j|.$, $\begin{array}{lclcr} {} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 0 \} &=& 4\mathbb{Z}, \\ {} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 1 \} &=& 1+4\mathbb{Z}, \\ {} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 2 \} &=& 2+4\mathbb{Z}, \\ {} &=& \{n\in\mathbb{Z} \mid n\bmod 4 = 3 \} &=& 3+4\mathbb{Z}. [a]} Let X be a finite set with n elements. Each part below gives a partition of $$A=\{a,b,c,d,e,f,g\}$$. The element in the brackets, [ ] is called the representative of the equivalence class. We have $$aRx$$ and $$xRb$$, so $$aRb$$ by transitivity. a The equivalence class of under the equivalence is the set of all elements of which are equivalent to. $$R= \{(a,a), (a,b),(b,a),(b,b),(c,c),(d,d)\}$$. That is, for all a, b and c in X: X together with the relation ~ is called a setoid. In general, if ∼ is an equivalence relation on a set X and x∈ X, the equivalence class of xconsists of all the elements of X which are equivalent to x. , is the quotient set of X by ~. equivalence class of a, denoted [a] and called the class of a for short, is the set of all elements x in A such that x is ... 8x 2A; x 2[a] ,xRa: When several equivalence relations on a set are under discussion, the notation [a] R is often used to denote the equivalence class of a under R. Theorem 1. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. X the class [x] is the inverse image of f(x). Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections that preserve partition structure. \{a,b,c\}} $$\therefore R$$ is reflexive. X $$ = \{...,-10,-6,-2,2,6,10,14,...\}$$ } For this relation $$\sim$$ on $$\mathbb{Z}$$ defined by $$m\sim n \,\Leftrightarrow\, 3\mid(m+2n)$$: a) show $$\sim$$ is an equivalence relation. { , For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. One may regard equivalence classes as objects with many aliases. Let be a set and be an equivalence relation on . a In each equivalence class, all the elements are related and every element in $$A$$ belongs to one and only one equivalence class. 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