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.
Answers
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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