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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 _______________ .
Options
0
1
`1/2`
-1
Solution
0
Let e be the identity element in Q \[-\] {1} with respect to * such that
\[a * e = a = e * a, \forall a \in Q - \left\{ 1 \right\}\]
\[a * e = a \text{ and }e * a = a, \forall a \in Q - \left\{ 1 \right\}\]
\[\text{ Then }, \]
\[a + e - ae = a \text{ and }e + a - ea = a, \forall a \in Q - \left\{ 1 \right\}\]
\[e\left( 1 - a \right) = 0 , \forall a \in Q - \left\{ 1 \right\}\]
\[e = 0 \in Q - \left\{ 1 \right\} \left[ \because a \neq 1 \right]\]
Thus, 0 is the identity element in Q \[-\] {1} with respect to *.
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