Question

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by , is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Answer 1

The given relation R in set A={ 1,2,3,4,5 } is defined as R={( a,b ):| a−b | is even}.

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

Let, ( a,b )∈R, thus | a−b | is even. Now, | −( a−b ) |=| b−a | is also even and ( b,a )∈R. Hence, Ris symmetric.

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

Therefore, the given relation R={( a,b ):| a−b | is even} in the set A={ 1,2,3,4,5 } is reflexive, symmetric and transitive and hence an equivalence relation.

Now, all the elements of the set { 1,2,3 } are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even.

Similarly, all elements of set { 2,4 } are related to each other as all the elements of this subset are even. Thus, the modulus of the difference between any two elements will be even.

Also, no element of the subset { 1,3,5 } is related to any element of the subset { 2,4 } as all the elements of { 1,3,5 }are odd while all the elements of { 2,4 } are even. Thus, the modulus of the difference between any two elements from each of these two subsets will not be even.