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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS2 Marks
Question
For each binary operation * defined below, determine whether * is commutative or associative.
On R - {-1}, define $\text{a}*\text{b}=\frac{\text{a}}{\text{b}+1}$
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Answer

For commutativity: $\text{a}*\text{b}=\frac{\text{a}}{\text{b}+1}\ \text{and}\ \text{b}*\text{a}=\frac{\text{b}}{\text{a}+1}\Rightarrow\ \ \text{a}*\text{b}\neq\text{b}*\text{a}$
For associativity: $\text{a}*(\text{b}*\text{c})=\text{a}*\Big(\frac{\text{b}}{\text{c}+1}\Big)=\frac{\text{a}}{\frac{\text{a}}{\text{c}+1}+1}=\frac{\text{a(c + a)}}{\text{b + c}+1}$
Also, $(\text{a}*\text{b})*\text{c}=\Big(\frac{\text{a}}{\text{b}+1}\Big)*\text{c}=\frac{\text{a}/\text{b}+1}{\text{c}+1/\text{c}}=\frac{\text{a}}{(\text{b}+1)(\text{c}+1)}$
$\therefore\ \ \text{a} * \text{(b} * \text{c)}\neq\text{(a} * \text{b)}* \text{c}$
Therefore, the operation * is neither commutative nor associative.

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