Let R 1 and R 2 be two relations defined on by a R 1 b a b 0 and a R 2 b a b . Then

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Solution

Q.
Let $R_{1}$ and $R_{2}$ be two relations defined on $ℝ$ by $a R_{1} b \Leftrightarrow a b \geq 0$ and $a R_{2} b \Leftrightarrow a \geq b .$ Then

1 $R_{1}$ is an equivalence relation but not $R_{2}$

2 $R_{2}$ is an equivalence relation but not $R_{1}$

3 Both $R_{1}$ and $R_{2}$ are equivalence relations

4 Neither $R_{1}$ nor $R_{2}$ is an equivalence relation

Ans.
(4)

$a = 1, b = 0, c = - 2 then transitive fails for R_{1}$

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