Question

On the set \(\mathbb{R}\) of real numbers, the relation \(\rho\) is defined by \(x \rho y,(x, y) \in \mathbb{R}\)
(A) If \(|x-y|<2\) then \(\rho\) is reflexive but neither symmetric nor transitive
(B) If \(x-y<2\) then \(\rho\) is reflexive and symmetric but not transitive
(C) If \(|x| \geq y\) then \(\rho\) is reflexive and transitive but not symmetric
(D) If \(x>|y|\) then \(\rho\) is transitive but neither reflexive nor symmetric

Answer 1

Ans: (D)
Hint : for option \(\mathrm{D}, \mathrm{x}>|\mathrm{x}|\) is not true hence not reflexive
Take \(x=2, y=-1\), clearly \(x>|y|\) but \(y>|x|\) doe not hold hence not symmetric
Now, Let \(x>|y|\) and \(y>|z| \Rightarrow x, y>0 . \therefore\) Rewriting, \(x>|y|\) and \(y>|z| \Rightarrow x>|z|\) hence transitive

Answer 2

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Answer 3

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