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

Question

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:

$0$
$1$
$\frac{1}2$
$-1$

✓

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, $\forall\text{ a}\in\text{Q}-\{-1\}$

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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