Show that the relation R defined on the set A = (1, 2, 3, 4, 5), given by
R = {(a, b) : |a – b| is even} is an equivalence relation.
In order to show R is an equivalence relation we need to show R is Reflexive, Symmetric and Transitive.
Given that, ∀ a, b ∈A, R = {(a, b) : |a – b| is even}.
Now,
R is Reflexive if (a,a) ∈ R ∀ a ∈ A
For any a ∈ A, we have
|a-a| = 0, which is even.
⇒ (a,a) ∈ R
Thus, R is reflexive.
R is Symmetric if (a,b) ∈ R ⇒ (b,a) ∈ R ∀ a,b ∈ A
(a,b) ∈ R
⇒ |a-b| is even.
⇒ |b-a| is even.
⇒ (b,a) ∈ R
Thus, R is symmetric .
R is Transitive if (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R ∀ a,b,c ∈ A
Let (a,b) ∈ R and (b,c) ∈ R ∀ a, b,c ∈ A
⇒ |a – b| is even and |b – c| is even
⇒ (a and b both are even or both odd) and (b and c both are even or both odd)
Now two cases arise:
Case 1 : when b is even
Let (a,b) ∈ R and (b,c) ∈ R
⇒ |a – b| is even and |b – c| is even
⇒ a is even and c is even [∵ b is even]
⇒ |a – c| is even [∵ difference of any two even natural numbers is even]
⇒ (a, c) ∈ R
Case 2 : when b is odd
Let (a,b) ∈ R and (b,c) ∈ R
⇒ |a – b| is even and |b – c| is even
⇒ a is odd and c is odd [∵ b is odd]
⇒ |a – c| is even [∵ difference of any two odd
natural numbers is even]
⇒ (a, c) ∈ R
Thus, R is transitive on Z.
Since R is reflexive, symmetric and transitive it is an equivalence relation on Z.