## Determine whether the relation R defined on the set R on all real numbers as R = {(a, b): a., b ∈R and a-b+√3 ∈ , where S is the

Question

Determine whether the relation R defined on the set R on all real numbers

as R = {(a, b): a., b ∈R and a-b+√3 ∈ , where S is the set of all irrational

numbers}, is reflexive, symmetric and transitive.

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Math
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2021-10-01T14:34:51+00:00
2021-10-01T14:34:51+00:00 1 Answer
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## Answers ( )

SOLUTIONTOCHECKThe relation R defined on the set R on 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

EVALUATIONHere the given relation is

R= {(a, b): a., b ∈ R and a-b+√3 ∈ S , where S is the set of all irrational numbers}CHECKINGFORREFLEXIVELet a ∈ R

Then a – a + √3 = √3

Which is an irrational number

Hence a – a + √3 ∈ S

So (a, a) ∈

RSois reflexive

RCHECKINGFORSYMMETRICLet (a , b) ∈

RThen a , b ∈ R and a-b+√3 ∈ S

⟹ – ( a-b+√3 ) ∈ S

⟹ – a + b – √3 ∈ S

⟹ b – a – √3 ∈ S

⟹ b – a – √3 + 2√3 ∈ S

⟹ b – a + √3 ∈ S

⟹ (b, a) ∈

RHenceis symmetric

RCHECKINGFORTRANSITIVELet (a , b) , ( b, c) ∈

RThen a , b, c ∈ R and a-b+√3 ∈ S & b-c+√3 ∈ S

⟹ ( a-c+2√3 ) ∈ S

⟹ a – c + √3 ∈ S

⟹ (a, c) ∈

RHence R is transitive

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