Find the identity element in the set of all rational numbers except −1 with respect to defined by a b = a + b + ab.

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Solution

Let e be the identity element inQ $-$ { $-$ 1} with respect to * such that
$a * e = a = e * a, \forall a \in Q - \{1\} a * e = a and e * a = a, \forall a \in Q - \{1\} a + e + a e = a and e + a + e a = a, \forall a \in Q - \{1\} e + a e = 0 and e + e a = 0, \forall a \in Q - \{1\} e (1 + a) = 0 and e (1 + a) = 0, \forall a \in Q - \{1\} e = 0, \forall a \in Q - \{- 1\} [∵ a ≠-1]$

Thus, 0 is the identity element in Q - {-1} with respect to *.

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