Question

Let $R$ be a relation on $R$, given by R=\\{(a, b): 3 a-3 b+\sqrt{7} is an irrational number \\}Then $R$ is

• A. reflexive and symmetric but not transitive
• B. reflexive and transitive but not symmetric
• C. reflexive but neither symmetric nor transitive
• D. an equivalence relation

Answer 1

Answer: (C)

reflexive but neither symmetric nor transitive

Answer 2

Check for reflexivity:
As 3(a−a)+7​=7​
which belongs to relation so relation is reflexive
Check for symmetric:
Take a=37​​,b=0
Now (a, b) ∈R but (b,a)∈/R
As 3(b−a)+7​=0
which is rational so relation is not symmetric.
Check for Transitivity:
Take (a, b) as (37​​,1)
&(b,c) as (1,327​​)
So now (a, b) ∈R&(b,c)∈R but (a,c)∈/R which means relation is not transitive