Question

Show that each of the relation R in the set , given by (i) (ii) is an equivalence relation. Find the set of all elements related to 1 in each case.

Answer 1

The given relation R in set A={ x∈Z:0≤x≤12 }={ 0,1,2,3,4,5,6,7,8,9,10,11,12 } is defined as

(i)

R={( a,b ):| a−b  | is a multiple of4}.

( a,a )∈R, since, for all values of a, a∈A, | a−a |=0 are multiple of 4. Hence, R is reflexive.

Let, ( a,b )∈R, thus | a−b  | is a multiple of 4. Also | −( a−b )  |=| b−a | is a multiple of 4. Then ( b,a )∈R. Hence, Ris symmetric.

Let, ( a,b ),( b,c )∈R, thus | a−b  |  and | b−c | is a multiple of 4. So, | a−c |=| ( a−b )+( b−c ) | is a multiple of 4. This implies that ( a,c )∈R, hence Ris transitive.

Therefore, the given relation R={( a,b ):| a−b  | is a multiple of4} in the set A={ x∈Z:0≤x≤12 }={ 0,1,2,3,4,5,6,7,8,9,10,11,12 } is reflexive, symmetric and transitive, and hence an equivalence relation.

The set of elements related to 1 is { 1,5,9 } since,

| 1−1 |=0 is a multiple of 4.

| 5−1 |=4 is a multiple of 4.

| 9−1 |=8 is a multiple of 4.

(ii)

R={ ( a,b ):a=b }

( a,a )∈R, since for all values of a, a∈A, a=a. Hence, R is reflexive.

Let, ( a,b )∈R, thus a=b, also b=a and ( b,a )∈R. Hence, Ris symmetric.

Let, ( a,b ),( b,c )∈R, thus a=b and b=c. So, a=cimplies ( a,c )∈R, hence Ris transitive.

Therefore, the given relation R={ ( a,b ):a=b } in the set A={ x∈Z:0≤x≤12 }={ 0,1,2,3,4,5,6,7,8,9,10,11,12 } is reflexive, symmetric and transitive and hence an equivalence relation.

The elements in R related to 1 are those elements in set A which are equal to 1. Hence, the set of elements related to 1 is { 1 }.