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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 + 3ϵ S, where S is the set of all irrational numbers}, is reflexive, symmetric and transitive.

                                                                                                OR

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

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


Solution

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

Reflexivity : Let a be any real number.       aa+3=3ϵS       (a,a)ϵR

So, R is reflexive.

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