Definition 5.14. A relation defined on a set is called an equivalencerelation if * is reflexive symmetricand transitive The usual notationfor an equivalencerelationis. Problem 5.15. Definea relation " on Z by (a, bye & if a and b are both even or both odd. Prove that is an equivalencerelation Definition 5.16. If ~ is an equivalencerelationon a set $ and x 6 5, then the equivalence class of *, written, or is given by the set () ES [ *my}- Exercise5.17. Whatare the equivalenceclasses in Exercise5.7? Problem5.157 Problem 5.18. Definca relation~ on IR by.X~} ifx -y is rational Prove that is an equivalencerelation Problem 5.19. Suppose w is an equivalencerelationon a sets. If a web for somea, be S, then E. = Es- Problem 5.20. Suppose is an equivalencerelationon a setA. Prove that ifs and f are elements of A, then either EnE, = 0 or E, =E. The punchline of Theorem 5.20 is that the equivalence classes of an equivalence relation partitionthe set A into pairwisedisjointsubsets Problem 5.21. Consider the relation ~ on E x (2\\ {}) defined by (a, b) (c, d) ifa.d = b. C. Prove that ~ is an equivalencerelation Question 5.22. Can you describe the equivalenceclasses in Problem 5.21 in terms of familiar mathematical objects? Problem 5.23. Let S = R x R and define a relation- on S by (a,b) ~(c,d) ifd-b = 4(c -@). Show that" is an equivalencerelation Question 5.24. Can you describe the equivalenceclasses in Problem 5.23 in terms of familiar mathematical objects? 5.2 Order relations Definition 5.25. Let ~ be a relation on a setX. Then ~ is said to be antisymmetriof for allxyEX, ifxy andy rex, then *=y. Definition 5.26. A relation