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 ∈ 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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