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

Answers ( )

    0
    2021-10-01T14:36:15+00:00

    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

    ━━━━━━━━━━━━━━━━

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