Choose the correct answer from the given four options.The identit… - Vidyadip

MCQ

Choose the correct answer from the given four options.The identity element for the binary operation $*$ defined on $\text{Q}\sim\{0\}$ as $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2}\ \forall\ \text{a, b}\in\text{Q}\sim\{0\}$ is:

A
$1$
B
$0$
✓
$2$
D
none of these.

✓

Answer

Correct option: C.

$2$

Given that, $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2}\ \forall\ \text{a, b}\in\text{Q}\sim\{0\}$
Let e be the identity element for $*$
$\therefore\ \text{a}\ ^*\ \text{e}=\frac{\text{ae}}{2}(\text{a}^*\ \text{e}=\text{e}^*\ \text{a}=\text{a})$
$\Rightarrow\ \text{a}=\frac{\text{ae}}{2}$
$\Rightarrow\ \text{e}=2$

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