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Question
Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.
Solution
Let e be the identity element in Q\[-\] {\[-\]1} with respect to * such that
\[a * e = a = e * a, \forall a \in Q - \left\{ 1 \right\}\]
\[a * e = a \text{ and }e * a = a, \forall a \in Q - \left\{ 1 \right\}\]
\[a + e + ae = a \text{ and }e + a + ea = a, \forall a \in Q - \left\{ 1 \right\}\]
\[e + ae = 0 \text{ and }e + ea = 0, \forall a \in Q - \left\{ 1 \right\}\]
\[e\left( 1 + a \right) = 0 \text{ and }e\left( 1 + a \right) = 0, \forall a \in Q - \left\{ 1 \right\}\]
\[e = 0, \forall a \in Q - \left\{ - 1 \right\} \text{ }\left[ \because a\neq-1 \right]\]
Thus, 0 is the identity element in Q - {-1} with respect to *.
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