Let * be a binary operation defined on set Q − 1 by the rule a * … - Vidyadip

Question

Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is:

$1$
$\frac{\text{a}-1}{\text{a}}$
$\frac{\text{a}}{\text{a}-1}$
$0$

✓

Answer

$0$

Solution:
Let e be the identity element in Q - {1} with respect to * such that
a * e = a = e * a, $\forall\text{ a}\in\text{Q}-\{-1\}$
a * e = a and e * a = a, $\forall\text{ a}\in\text{Q}-\{-1\}$
a + e - ae = a and e + a - ea = a, $\forall\text{ a}\in\text{Q}-\{-1\}$
e(1 - a) = 0, $\forall\text{ a}\in\text{Q}-\{-1\}$ $[\because \text{a}\neq1]$
Thus, 0 is the identity element in Q - {1} with respect to *.

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