## 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.

in progress 0
3 weeks 2021-10-01T14:34:51+00:00 1 Answer 0 views 0

1. ### SOLUTION

TO CHECK

The 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

EVALUATION

Here the given relation is

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

CHECKING FOR REFLEXIVE

Let a ∈ R

Then a – a + √3 = √3

Which is an irrational number

Hence a – a + √3 ∈ S

So (a, a) ∈ R

So R is reflexive

CHECKING FOR SYMMETRIC

Let (a , b) ∈ R

Then 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) ∈ R

Hence R is symmetric

CHECKING FOR TRANSITIVE

Let (a , b) , ( b, c) ∈ R

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

⟹ ( a-c+2√3 ) ∈ S

⟹ a – c + √3 ∈ S

⟹ (a, c) ∈ R

Hence R is transitive

━━━━━━━━━━━━━━━━ Show that the set Q+ of all positive rational numbers forms an abelian group under the operation * defined by