ii No element of E is related to any element of O and vice versa iii E and O are disjoint and Z EUO The subset E is called the equivalence class containing zero and is denoted by 0 Similarly O is the equivalence class containing 1 and is denoted by 1 Note that 0 1 0 2r and 1 2r 1 r Z Infact what we have seen above is true for an arbitrary equivalence relation R in a set X Given an arbitrary equivalence relation R in an arbitrary set X R divides X into mutually disjoint subsets A called partitions or subdivisions of X satisfying i all elements of A are related to each other for all i ii no element of A is related to any element of A iii UA X and A A 0 i j The subsets A are called equivalence classes The interesting part of the situation is that we can go reverse also For example consider a subdivision of the set Z given by three mutually disjoint subsets A A and A hose union is Z with A x Z x is a multiple of 3 6 3 0 3 6 A x Z x 1 is a multiple of 3 5 2 1 4 7 A x Z x 2 is a multiple of 3 4 1 2 5 8 Define a relation R in Z given by R a b 3 divides a b Following the arguments similar to those used in Example 5 we can show that R is an equivalence relation Also A coincides with the set of all integers in Z which are related to zero A coincides with the set of all integers which are related to 1 and A coincides with the set of all integers in Z which are related 2 Thus A 0 A 1 and A 2 In fact A 3r A 3r 1 and A 3r 2 for all r Z Example 6 Let R be the relation defined in the set A 1 2 3 4 5 6 7 by R a b both a and b are either odd or even Show that R is an equivalence relation Further show that all the elements of the subset 1 3 5 7 are related to each other and all the elements of the subset 2 4 6 are related to each other but no element of the subset 1 3 5 7 is related to any element of the subset 2 4 6 Rationalised 2023 24 RELATIONS AND FUNCTIONS E Solution Given any element a in A both a and a must be either odd or even so that a a e R Further a b R both a and b must be either odd or even b a R Similarly a b R and b c R all elements a b c must be either even or odd simultaneously a c R Hence R is an equivalence relation Further all the elements of 1 3 5 7 are related to each other as all the elements of this subset are odd Similarly all the elements of the subset 2 4 6 are related to each other as all of them are even Also no element of the subset 1 3 5 7 can be related to any element of 2 4 6 as elements of 1 3 5 7 are odd while elements of 2 4 6 are even EXERCISE 1 1 1 Determine whether each of the following relations are reflexive syr transitive 5 1 Relation R in the set A 1 2 3 13 14 defined R x y 3x y 0 ii Relation R in the set N of natural numbers defined as