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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS2 Marks
Question
Examine whether the operation $^* $defined on $R$ by $\text{a}^*\text{b}=\text{ab}+1$ is $(i)$ a binary or not. $(ii)$ if a binary operation, is it associative or not?
✓

Answer

The given operation is $\text{a}^*\text{b}=\text{ab}+1$
If any operation is a binary operation then it must follow the closure property.
Let $\text{a}\in\text{R},\text{b}\in\text{R}$
then $\text{a}^*\text{b}\in\text{R}$
also $\text{ab}+1\in\text{R}$
i.e. $\text{a}^*\text{b}\in\text{R}$
So $^*$ on $R$ satisfies the closure property
Now if this binary operation satisfies associative law then
$(\text{a}^*\text{b})^*\text{c}=\text{a}^*(\text{b}^*\text{c})$
$(\text{a}^*\text{b})^*\text{c}=(\text{ab}+1)^*\text{c}$
$=(\text{ab}+1)\text{c}+1$
$=\text{abc}+\text{c}+1$
$\text{a}^*(\text{b}^*\text{c})=\text{a}^*(\text{bc}+1)$
$=\text{a}(\text{bc}+1)+1$
$=\text{abc}+\text{a}+1$
$\therefore(\text{a}^*\text{b})^*\text{c}\neq\text{a}^*(\text{b}^*\text{c})$
i.e., $^*$ operation does not follow associative law.

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