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

Question

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:

1
0
2
none of these.

✓

Answer

2

Solution:
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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