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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations4 Marks
Question
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.
✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}.$ Then,

$\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$

$=\frac{\text{ba}}{4}$

$=\text{b}\ ^*\ \text{a}$

Therefore,

$\text{a}\ ^*\ \text{b}=\text{b}\ ^*\ \text{a},\ \forall\ \text{a, b}\in\text{Q}$

Thus '*' is commutative on Z.

Associativity: Let $\text{a, b, c}\in\text{Q}.$ Then,

$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \Big(\frac{\text{bc}}{4}\Big)$

$=\frac{\text{a}\big(\frac{\text{bc}}{4}\big)}{4}$

$=\frac{\text{abc}}{16}$

$(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}=\frac{\text{ab}}{4}\ ^*\ \text{c}$

$=\frac{\big(\frac{\text{ab}}{4}\big)\text{c}}{4}$

$=\frac{\text{abc}}{16}$

Therefore,

$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=(\text{a}\ ^*\ \text{b})\ ^*\ \text{c},\ \forall\ \text{a, b, c}\in\text{Q}$

Thus, '*' is associative on Q.

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