defined in the set A = { , , , , , , } 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 { , , , } are related to each other and all the elements of the subset { , , } are related to each other, but no element of the subset { , , , } is related to any element of the subset { , , }. Solution Given any element a in A, both a and a must be either odd or even, so that ( a , a ) ∈ 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 { , , , } are related to each other, as all the elements of this subset are odd.
Similarly, all the elements of the subset { , , } are related to each other, as all of them are even. Also, no element of the subset { , , , } can be related to any element of { , , }, as elements of { , , , } are odd, while elements of { , , } are even. EXERCISE . .
Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = { , , , ..., , } defined as R = {( x , y ) : x – y = } (ii) Relation R in the set N of natural numbers defined as R = {( x , y ) : y = x + and x < } (iii) Relation R