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Relation — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRelation2 Marks
Question
For the relation $R_1$ defined on R by the rule $(\text{a, b})\in\text{R}_1\Leftrightarrow1+\text{ab}>0$ Prove that, $(\text{a, b})\in\text{R}_1$ and $(\text{a},\text{b})\in\text{R}_1$ and $(\text{b},\text{c})\in\text{R}_1\Rightarrow(\text{a, c})\in\text{R}_1$ is not true for all $\text{a, b, c}\in\text{R}$
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Answer

Let $\Big(1,\frac{-1}{2}\Big)\in\text{R}_1$ and $\Big(\frac{-1}{2},-4\Big)\in\text{R}_1$ $\Rightarrow1+1\times\frac{-1}{2}>0$ and $1+\Big(\frac{-1}{2}\Big)-4>0$ But, $1+1\times(-4)=1-4$ $=-3<0$ So, $(1,-4)\notin\text{R}_1$

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