Find the identity element in the set of all rational numbers except �

1.	Find the identity element in the set of all rational numbers except – 1 with respect to * defined by a * b = a + b + ab.
Answer» Let e be the identity element in I⁺ with respect to * Such that, a * e = a = e * a, ∀ a ∈ Q – {-1} a * e = a and e * a = a, ∀ a ∈ Q – {-1} a + e + ae = a and e + a + ea = a, ∀ a ∈ Q – {-1} e + ae = 0 and e + ea = 0, ∀ a ∈ Q – {-1} e(1 + a) = 0 and e(1 + a) = 0, ∀ a ∈ Q – {-1} e = 0, ∀ a ∈ Q – {-1} [because a ! = -1] Hence, 0 is the identity element in Q – {-1} with respect to *.

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

Answer»

Let e be the identity element in I⁺ with respect to *

Such that, a * e = a = e * a, ∀ a ∈ Q – {-1}

a * e = a and e * a = a, ∀ a ∈ Q – {-1}

a + e + ae = a and e + a + ea = a, ∀ a ∈ Q – {-1}

e + ae = 0 and e + ea = 0, ∀ a ∈ Q – {-1}

e(1 + a) = 0 and e(1 + a) = 0, ∀ a ∈ Q – {-1}

e = 0, ∀ a ∈ Q – {-1} [because a ! = -1]

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