Question

Determine whether the relation R defined on the set R of all real numbers as R = {(a, b) ; a, b $ϵ$ R and a-b + $\sqrt{3}ϵ$ S, where S is the set of all irrational numbers}, is reflexive, symmetric and transitive.

OR

Let A = $R\times R$ and $\ast$ be the binary operation on A defined by (a,b) $\ast$ (c, d) - (a + c, b + d).

Prove that $\ast$ is commutative and associative. Find the identity element for $\ast$ on A. Also write the inverse element of the element (3, -5) in A.

Answer 1

We have R = {(a, b) : a, b $ϵ$ R and a - b + $\sqrt{3}ϵ$ S is the set of all irrational numbers}

Reflexivity : Let a be any real number. $\therefore a-a+\sqrt{3}=\sqrt{3}ϵS\therefore (a,a)ϵR$

So, R is reflexive.