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1.Relation and Function
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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
A
$R_{1}$ is an equivalence relation but not $R_{2}$
B
$R_{2}$ is an equivalence relation but not $R_{1}$
C
both $R_{1}$ and $R_{2}$ are equivalence relations
D
neither $R_{1}$ nor $R_{2}$ is an equivalence relation
(JEE MAIN-2022)
Solution
$R_{1}=\{x y \geq 0, x, y \in R\}$
For reflexive $x \times x \geq 0$ which is true.
For symmetric
If $x y \geq 0 \Rightarrow y x \geq 0$
If $x =2, y =0$ and $z =-2$
Then $x . y \geq 0 \& y . z \geq 0$ but $x . z \geq 0$ is not true $\Rightarrow$ not transitive relation.
$R_{ I }$ is not equivalence
$R _{2}$ if $a \geq b$ it does not implies $b \geq a$
$R_{2}$ is not equivalence relation
$D$
Standard 12
Mathematics
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