EXAMPLE 33. 9 Equivalence Relations In the study of mathematics, we deal with many examples of relations be-tween elements of various sets. 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 ˘. De ne the relation R on A by xRy if xR 1 y and xR 2 y. 2. Solution. Prove that the binary operation + on R=Z given by a+ b= a+ b is well-de ned. 1 is an equivalence relation on A. 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. The order of the elements in a set doesn't contribute Therefore ~ is an equivalence relation because ~ is the kernel relation of x = x. (I will omit the proof that R=Z is a group.) Selected solutions to problems Problem Set 2 2.De ne a relation ˘on R given by a˘bif a b2Z. 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. Examples of Other Equivalence Relations. Proof. Give the rst two steps of the proof that R is an equivalence relation by showing that R is re exive and symmetric. Obviously, then, we will have that: 1. If x = y, then y = x. Example 6. That is, ”x 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. Let Rbe the relation on Z de ned by aRbif a+3b2E. 1. Symmetric. 1. 2 are equivalence relations on a set A. Reﬂexive. The relation \(\sim\) on \(\mathbb{Q}\) from Progress Check 7.9 is an equivalence relation. 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. The equivalence classes of this relation are the orbits of a group action. The set [x] ˘as de ned in the proof of Theorem 1 is called the equivalence class, or simply class of x under ˘. Equivalence Relations • A relation on a set that satisfies the three properties of reflexivity, symmetry, and transitivity is called an equivalence relation. Proof. Equivalence Relations De nition 2.1. 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. There is an equivalence relation which respects the essential properties of some class of problems. 4 CS 441 Discrete mathematics for CS M. Hauskrecht Equality Definition: Two sets are equal if and only if they have the same elements. We write X= ˘= f[x] ˘jx 2Xg. For example, in working with the integers, we encounter relations such as ”x is less than y”. De nition 4. 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. Let Xbe a set. Equivalence relations A motivating example for equivalence relations is the problem of con-structing the rational numbers. Notice the importance of the ordering of the elements of the set in this relation. But di erent ordered … (c)Is (R=Z;+) a group? 2. Let ˘be an equivalence relation on X. If we consider the equivalence relation as de ned in Example 5, we have two equiva-lence … 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: Equality of real numbers is another example of an equivalence relation. 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). (a)Prove that ˘is an equivalence relation. 2. 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. Re exive: Let a 2A. By one of the above examples, Ris an equivalence relation.
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