Let R_1 and R_2 be two relations defined on R by aR_1b ab 0 and aR_2b a b. Then,

R_1=\xy 0, x, y R\ For reflexive x x 0 which is true. For symmetric If x y 0 y x 0 If x=2, y=0 and z=-2 Then x y 0 \& y&nbsp

Let R_{1} and R_{2} be two relations defined on R by aR_{1}b rightarrow ab ? 0 and aR_{2}b rightarrow a ? b. Then,: Relation (Mathematics)

Exam ( $27^{th\ }$ July $1^{st\ }$ Shift 2022)

Mathematics

Let

R_{1}

and

R_{2}

be two relations defined on

R

aR_{1}b \Leftrightarrow ab \geq 0

and

aR_{2}b \Leftrightarrow a \geq b

. Then,

R_{1}

is an equivalence relation but not

R_{2}

R_{2}

is an equivalence relation but not

R_{1}

both

R_{1}

and

R_{2}

are equivalence relations

neither

R_{1}

nor

R_{2}

is an equivalence relation

Correct Answer:

$\mathrm{R}_1=\{\mathrm{xy} \geq 0, \mathrm{x}, \mathrm{y} \in \mathrm{R}\}$

For reflexive $\mathrm{x} \times \mathrm{x} \geq 0$ which is true.

For symmetric

If $x y \geq 0 \Rightarrow y x \geq 0$

If $\mathrm{x}=2, \mathrm{y}=0$ and $\mathrm{z}=-2$

Then $x \cdot y \geq 0$ \& $y \cdot z \geq 0$ but $x \cdot z \geq 0$ is not true

⇒ not transitive relation.

$\Rightarrow R_1$ is not equivalence

$\mathrm{R}_2$ if $\mathrm{a} \geq \mathrm{b}$ it does not implies $\mathrm{b} \geq \mathrm{a}$

$\Rightarrow R_2$ is not equivalence relation

⇒ D

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