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