If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence … 2 are equivalence relations on a set A. Proof. (a)Prove that ˘is an equivalence relation. Re exive: Let a 2A. Reflexive. EXAMPLE 33. For example, in working with the integers, we encounter relations such as ”x is less than y”. An equivalence relation on a set is a relation with a certain combination of properties that allow us to sort the elements of the set into certain classes. Proof. Then ~ is an equivalence relation because it is the kernel relation of function f:S N defined by f(x) = x mod n. Example: Let x~y iff x+y is even over Z. 2. Here the equivalence relation is called row equivalence by most authors; we call it left equivalence. Note that x+y is even iff x and y are both even or both odd iff x mod 2 = y mod 2. Prove that the binary operation + on R=Z given by a+ b= a+ b is well-de ned. The order of the elements in a set doesn't contribute Let Rbe the relation on Z de ned by aRbif a+3b2E. Here, rather than working with triangles we work with numbers: we say that the real numbers x and y are equivalent if we simply have that x = y. Solution. Equivalence Relations De nition 2.1. Then since R 1 and R 2 are re exive, aR 1 a and aR 2 a, so aRa and R is re exive. (b)Let R=Z denote the set of equivalence classes of ˘. The set [x] ˘as de ned in the proof of Theorem 1 is called the equivalence class, or simply class of x under ˘. De ne the relation R on A by xRy if xR 1 y and xR 2 y. 9 Equivalence Relations In the study of mathematics, we deal with many examples of relations be-tween elements of various sets. That is, ”x less than y” We write X= ˘= f[x] ˘jx 2Xg. (I will omit the proof that R=Z is a group.) There is an equivalence relation which respects the essential properties of some class of problems. Symmetric. Example: The relation R on a set {1,2,3,4}, and a relation R defined over X as (x,y) ∈ R if x <= y: But di erent ordered … 4 CS 441 Discrete mathematics for CS M. Hauskrecht Equality Definition: Two sets are equal if and only if they have the same elements. 2. Let Xbe a set. Example: • {1,2,3} = {3,1,2} = {1,2,1,3,2} Note: Duplicates don't contribute anythi ng new to a set, so remove them. Examples of Other Equivalence Relations. A rational number is the same thing as a fraction a=b, a;b2Z and b6= 0, and hence speci ed by the pair ( a;b) 2 Z (Zf 0g). 1. De nition 4. Equivalence Relations • A relation on a set that satisfies the three properties of reflexivity, symmetry, and transitivity is called an equivalence relation. 1 is an equivalence relation on A. Therefore ~ is an equivalence relation because ~ is the kernel relation of By one of the above examples, Ris an equivalence relation. Obviously, then, we will have that: 1. 2. If x = y, then y = x. The relation \(\sim\) on \(\mathbb{Q}\) from Progress Check 7.9 is an equivalence relation. Example 6. Conclusion: Theorems 31 and 32 imply that there is a bijection between the set of all equivalence relations of Aand the set of all partitions on A. For example, if X= I2 is the unit square, glueing together opposite ends of X(with the same orientation) ‘should’ produce the torus S 1 S. To encapsulate the (set-theoretic) idea of glueing, let us recall the de nition of an equivalence relation on a set. x = x. Notice the importance of the ordering of the elements of the set in this relation. (c)Is (R=Z;+) a group? The equivalence classes of this relation are the orbits of a group action. 1. Let ˘be an equivalence relation on X. Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric. Selected solutions to problems Problem Set 2 2.De ne a relation ˘on R given by a˘bif a b2Z. 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