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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
The binary operation * is defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{7}$ on the set Q of all rational numbers. Show that * is associative.
✓

Answer

The binary operator * is defined as,
$\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{7}$ for all $\text{a, b}\in\text{Q}$
Now,
Associativity: Let $\text{a, b, c}\in\text{Q},$ then
$(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}=\frac{\text{ab}}{7}\ ^*\ \text{c}=\frac{\text{abc}}{49}\ ....(\text{i})$
and $\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \frac{\text{bc}}{7}=\frac{\text{abc}}{49}\ .....(\text{ii})$
From (i) and (ii)
(a * b) * c = a * (b * c)
⇒ '*' is associative on Q.

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