Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, 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}.

It is given that the relation R in the set A = {1, 2, 3, 4, 5}

given by R = {(a, b) : |a – b| is even},

It is clear that for any element a ϵ A, we have |a – a|= 0(which is even)

Therefore, R is reflexive.

Now, Let (a,b) ϵ R

⇒ |a – b| is even

⇒ |-(a – b)| = |b - a| is also even.

⇒ (b,a) ϵ R

Therefore, R is symmetric.

Now, Let (a,b) ϵ R and (b,c) ϵ R .

⇒ |a – b| is even and |b – c| is even

⇒ (a – b) is even and (b – c) is even

⇒ (a – c) = (a – b) + (b – c) is even

⇒ |a – c| is even

⇒ (a,c) ϵ R

Therefore, R is transitive.

Therefore, R is an equivalence relation.

Now, all elements of the set {1,3,5} are related to each other as all the elements of this subset are odd.

Therefore, the modules of the difference between any two elements will be even.

Similarly, all elements of the sets {2,4} are related each other as all the elements of this subset are even.

Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2,4} are even.

Therefore, the modulus of the difference between the two elements will not be even.