Question

Let the relation \(\rho\) be defined on \(\mathbb{R}\) by a \(\rho b\) holds if and only if \(a-b\) is zero or irrational, then
(A) \(\rho\) is equivalence relation
(B) \(\rho\) is reflexive \& symmetric but is not transitive
(C) \(\rho\) is reflexive and transitive but is not symmetric
(D) \(\rho\) is reflexive only

Answer 1

Ans : (B)
Hint : If \(a-b=0\) then \(b-a=0\),
if \(a-b\) is irrational then \(b-a\) is irrational
\(\therefore a \rho b \Rightarrow b \rho a \Rightarrow\) symmetric
\(\forall a \in \mathbb{R}, a-a=0 a \rho a \Rightarrow\) reflexive
If \(a=2, b=\sqrt{2}, c=3\), then
\(a \rho b, b \rho c\) but \(a \rho c\) is not true \(\Rightarrow\) not transitive

Answer 2

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Answer 3

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