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

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