So \(a\ M\ b\) if and only if there exists a \(k \in \mathbb{Z}\) such that \(a = bk\). Then \(a \equiv b\) (mod \(n\)) if and only if \(a\) and \(b\) have the same remainder when divided by \(n\). For example, "is greater than," "is at least as great as," and "is equal to" (equality) are transitive relations: 1. whenever A > B and B > C, then also A > C 2. whenever A ≥ B and B ≥ C, then also A ≥ C 3. whenever A = B and B = C, then also A = C. On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. In doing this, we are saying that the cans of one type of soft drink are equivalent, and we are using the mathematical notion of an equivalence relation. Definitions A relation that is reflexive, symmetric, and transitive on a set S is called an equivalence relation on S. That is, the ordered pair \((A, B)\) is in the relaiton \(\sim\) if and only if \(A\) and \(B\) are disjoint. We have now proven that \(\sim\) is an equivalence relation on \(\mathbb{R}\). Theorem 2. We know this equality relation on \(\mathbb{Z}\) has the following properties: In mathematics, when something satisfies certain properties, we often ask if other things satisfy the same properties. This is called Antisymmetric Relation. The most familiar (and important) example of an equivalence relation is identity . Examples: Let S = ℤ and define R = {(x,y) | x and y have the same parity} i.e., x and y are either both even or both odd. For \(a, b \in A\), if \(\sim\) is an equivalence relation on \(A\) and \(a\) \(\sim\) \(b\), we say that \(a\) is equivalent to \(b\). Hence, since \(b \equiv r\) (mod \(n\)), we can conclude that \(r \equiv b\) (mod \(n\)). Justify all conclusions. Leibinz equality eq is an equivalence relation. Then \(0 \le r < n\) and, by Theorem 3.31, Now, using the facts that \(a \equiv b\) (mod \(n\)) and \(b \equiv r\) (mod \(n\)), we can use the transitive property to conclude that, This means that there exists an integer \(q\) such that \(a - r = nq\) or that. For\(l_1, l_2 \in \mathcal{L}\), \(l_1\ P\ l_2\) if and only if \(l_1\) is parallel to \(l_2\) or \(l_1 = l_2\). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on"). Is the relation R antisymmetric? The subset relation is reflexive, antisymmetric, and transitive. That is, \(\mathcal{P}(U)\) is the set of all subsets of \(U\). Is the relation \(T\) symmetric? \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:tsundstrom2", "Equivalence Relations", "congruence modulo\u00a0n" ], 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%2F7.2%253A_Equivalence_Relations, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), ScholarWorks @Grand Valley State University, Directed Graphs and Properties of Relations. Theorem 2. Let \(U\) be a nonempty set and let \(\mathcal{P}(U)\) be the power set of \(U\). Equality is the model of equivalence relations, but some other examples are: Equality mod m: The relation x = y (mod m) that holds when x and y have the same remainder when divided by m is an equivalence relation. Explain why congruence modulo n is a relation on \(\mathbb{Z}\). Define the relation \(\sim\) on \(\mathbb{Q}\) as follows: For all \(a, b \in Q\), \(a\) \(\sim\) \(b\) if and only if \(a - b \in \mathbb{Z}\). Since \(0 \in \mathbb{Z}\), we conclude that \(a\) \(\sim\) \(a\). Draw a directed graph of a relation on \(A\) that is antisymmetric and draw a directed graph of a relation on \(A\) that is not antisymmetric. If \(a \sim b\), then there exists an integer \(k\) such that \(a - b = 2k\pi\) and, hence, \(a = b + k(2\pi)\). As the following exercise shows, the set of equivalences classes may be very large indeed. In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. End Leibniz. REFLEXIVE, SYMMETRIC and TRANSITIVE RELATIONS© Copyright 2017, Neha Agrawal. That is, if \(a\ R\ b\) and \(b\ R\ c\), then \(a\ R\ c\). Thus, according to Theorem 8.3.1, the relation induced by a partition is an equivalence relation. That is, a is congruent modulo n to its remainder \(r\) when it is divided by \(n\). Do not delete this text first. It is now time to look at some other type of examples, which may prove to be more interesting. \end{array}\]. Let \(A\) be nonempty set and let \(R\) be a relation on \(A\). If \(x\ R\ y\), then \(y\ R\ x\) since \(R\) is symmetric. Being the same size as is an equivalence relation; so are being in the same row as and having the same parents as. Now, \(x\ R\ y\) and \(y\ R\ x\), and since \(R\) is transitive, we can conclude that \(x\ R\ x\). I need a little help on this. So let \(A\) be a nonempty set and let \(R\) be a relation on \(A\). The parity relation is an equivalence relation. (e) Carefully explain what it means to say that a relation on a set \(A\) is not antisymmetric. In these examples, keep in mind that there is a subtle difference between the reflexive property and the other two properties. This list of fathers and sons and how they are related on the guest list is actually mathematical! In this section, we focused on the properties of a relation that are part of the definition of an equivalence relation. Corollary. Let \(x, y \in A\). Equivalence relations. We give an equivalent definition, up-to an equivalence relation on the carrier. Let \(a, b \in \mathbb{Z}\) and let \(n \in \mathbb{N}\). When we choose a particular can of one type of soft drink, we are assuming that all the cans are essentially the same. Asymmetric. For example, when you go to a store to buy a cold soft drink, the cans of soft drinks in the cooler are often sorted by brand and type of soft drink. For each \(a \in \mathbb{Z}\), \(a = b\) and so \(a\ R\ a\). … In this context, antisymmetry means that the only way each of two numbers can be divisible by the other is if the two are, in fact, the same number; equivalently, if n and m are distinct and n is a factor of m, then m cannot be a factor of n. For example, 12 is divisible by 4, but 4 is not divisible by 12. Assume that \(a \equiv b\) (mod \(n\)), and let \(r\) be the least nonnegative remainder when \(b\) is divided by \(n\). As long as no two people pay each other's bills, the relation is antisymmetric. Let us assume that R be a relation on the set of ordered pairs of positive integers such that ((a, b), (c, d))∈ R if and only if ad=bc. Examples: Let S = ℤ and define R = {(x,y) | x and y have the same parity} i.e., x and y are either both even or both odd. 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