First slide

Relations XII

Question

For $x, y \in R$ , define a relation $R$ by $x R y$
if and only if $x - y + \sqrt{2}$ is an irrational number. Then $R$ is

Moderate

A

reflexive
B

symmetric
C

transitive
D

none of these

Solution

Since $x - x + \sqrt{2} = \sqrt{2}$ which is an irrational number so $x R x$ for all $x \in R .$ Hence $R$ is reflexive. $R$ is
not symmetric as $(\sqrt{2}, 1) \in R$ but $(1, \sqrt{2}) \notin R$ Again $R$
is not transitive since $(\sqrt{2}, 1) \in R$ and $(1, 2 \sqrt{2}) \in R$ but $(\sqrt{2}, 2 \sqrt{2}) \notin R$ .

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