Let us see a few more examples of equivalence relations. Just to give an example of a relation, let’s take the family P(A) of subsets of the set A= fb;cg: Equivalence relations. De nition 3. Examples. Let X =Z, fix 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. R is an equivalence relation on A if R is reflexive, symmetric, and transitive. For each 1 m 7 find all pairs 5 x;y 10 such that x y(m). Example 32. 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. 2 Modular Arithmetic The most important reason that we are thinking about equivalence relations is to apply them to a particular situation. Here the equivalence relation is called row equivalence by most authors; we call it left equivalence. The relation is symmetric but not transitive. Proof. We can then write Z= ˘= ffodd integersg, feven integersgg. Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric. Example Which of the following relations are reflexive, where each is defined Then ~ is an equivalence relation with equivalence classes [0]=evens, and [1]=odds. Relations are somewhat general, and don’t 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. The equivalence classes of this relation are the orbits of a group action. Example. De ne the relation R on A by xRy if xR 1 y and xR 2 y. It was a homework problem. Exercise 33. • The equivalence class of (2,3): [(2,3)] = {(2k,3k)|k ∈ Z+}. Exercise 34. 3. • R is an equivalence relation. • 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 Show that congruence mod m is an equivalence relation (the only non-trivial part is 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 In that case we write a b(m). 2 are equivalence relations on a set A. Proof. Re exive: Let a 2A. Problem 3. The relations “has the same hair color as” or “is the same age as” in the set of people are equivalence relations. Let Rbe a relation de ned on the set Z by aRbif a6= b. This is true. Then Ris symmetric and transitive. Then R is an equivalence relation on X if it satis es the following properties. Let X be a set and let R X X. Example 3.7.1. Problem 2. 1. One of which I am fond is a \partial order relation", like \is a subset of" among subsets of a given set. Properties of Relations Definition A relation R : A !A is said to be reflexive if xRx for all x 2A. For every equivalence relation R, the function nat(R): A Æ A/R mapping every element x Œ A onto [[x]] is called a natural mapping of A onto A/R. EQUIVALENCE RELATIONS 38 3.7. Then since R 1 and R 2 are re exive, aR 1 a and aR 2 a, so aRa and R is re exive. 4.1 Example 1 This example comes from number theory: fix a non-zero integer d. We say The intersection of two equivalence relations on a nonempty set A is an equivalence relation. Section 3: Equivalence Relations • Definition: Let R be a binary relation on A . 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. 3. This is false. Proof. If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence classes: odds and evens. Example 6. 2. 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