Consider the binary operation * defined on Q − 1 by the rule a^ *… - Vidyadip

MCQ

Consider the binary operation $*$ defined on $Q − \{1\}$ by the rule $a^ * b = a + b − ab$ for all $a, b \in Q − \{1\}.$ The identity element in $Q − \{1\}$ is:

✓
$0$
B
$1$
C
$\frac{1}2$
D
$-1$

✓

Answer

Correct option: A.

$0$

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

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