Question

Determine whether the relation R defined on the set R of all real numbers as R = {(a, b) : a, b ∈ R and a – b + 3–√ ∈ S, where S is the set of all irrational numbers}, is reflexive, symmetric and transitive.

Answer 1

Here, relation R defined on the set R is given as

Reflexivity : Let a ∈ R (set of real numbers)

Now, (a,a) ∈ R as a - a + \(\sqrt{3}\) = \(\sqrt{3}\) ∈ S

i.e., R is reflexive ...(i)

Symmetric : Let a,b ∈ R (set of real numbers)

i.e., R is symmetric ...(ii)

Transitivity : Let a,b,c ∈ R

i.e., R is transitive. ...(iii)

(i),(ii) and (iii) ⇒ R is reflexive, symmetric and transitive