Example 6. Just to give an example of a relation, letâs take the family P(A) of subsets of the set A= fb;cg: For every equivalence relation R, the function nat(R): A Æ A/R mapping every element x Å A onto [[x]] is called a natural mapping of A onto A/R. Let us see a few more examples of equivalence relations. Example. EQUIVALENCE RELATIONS 38 3.7. 3. In that case we write a b(m). 2. Let X =Z, ï¬x m 1 and say a;b 2X are congruent mod m if mja b, that is if there is q 2Z such that a b =mq. 2 are equivalence relations on a set A. This is false. The relations âhas the same hair color asâ or âis the same age asâ in the set of people are equivalence relations. 4.1 Example 1 This example comes from number theory: ï¬x a non-zero integer d. We say It was a homework problem. Exercise 34. presently interested in, namely an equivalence relation, but there are other kinds of relations. There is an equivalence relation which respects the essential properties of some class of problems. Equivalence relations. Let R be the relation on the set of ordered pairs of positive inte-gers such that (a,b)R(c,d) if and only if ad = bc. Re exive: Let a 2A. This is true. Example 3.7.1. De nition 3. The relation is symmetric but not transitive. Example Which of the following relations are reï¬exive, where each is deï¬ned Problem 2. Relations are somewhat general, and donât say very much about sets; therefore, we intro-duce the concept of the equivalence relation, which is a slightly more speci cally-de ned relation. Proof. De ne the relation R on A by xRy if xR 1 y and xR 2 y. Re exivity: For every x 2X, (x;x) 2R: Then ~ is an equivalence relation with equivalence classes [0]=evens, and [1]=odds. Here the equivalence relation is called row equivalence by most authors; we call it left equivalence. Proof. All the proofs will make use of the â¼ deï¬nition above: 1The notation U ×U means the set of all ordered pairs ( x,y), where belong to U. 1. One of which I am fond is a \partial order relation", like \is a subset of" among subsets of a given set. â¢ The equivalence class of (2,3): [(2,3)] = {(2k,3k)|k â Z+}. Examples. For each 1 m 7 ï¬nd all pairs 5 x;y 10 such that x y(m). â¢ R is an equivalence relation. R is an equivalence relation on A if R is reflexive, symmetric, and transitive. Let Rbe a relation de ned on the set Z by aRbif a6= b. The intersection of two equivalence relations on a nonempty set A is an equivalence relation. Problem 3. We can then write Z= Ë= ffodd integersg, feven integersgg. Proof. 3. Section 3: Equivalence Relations â¢ Definition: Let R be a binary relation on A . If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence classes: odds and evens. Then R is an equivalence relation on X if it satis es the following properties. â¢ From the last section, we demonstrated that Equality on the Real Numbers and Congruence Modulo p on the Integers were reflexive, symmetric, and transitive, so we can describe 2 Modular Arithmetic The most important reason that we are thinking about equivalence relations is to apply them to a particular situation. Exercise 33. Note that {[0],[1]} is a partition of Z. CS340-Discrete Structures Section 4.2 Page 25 Equivalence Classes Example: The set of real numbers R can be partitioned into the set of Let X be a set and let R X X. Properties of Relations Deï¬nition A relation R : A !A is said to be reï¬exive if xRx for all x 2A. Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric. Show that congruence mod m is an equivalence relation (the only non-trivial part is The equivalence classes of this relation are the orbits of a group action. 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