Consider a binary operation on Q – {1}, defined by a * b = a + b - a

1.	Consider a binary operation on Q – {1}, defined by a * b = a + b - ab. (i) Find the identity element in Q – {1}. (ii) Show that each a ∈ Q - {1} has its inverse.
Answer» (i) For a binary operation , e identity element exists if ae = ea = a. As ab = a+ b- ab ae = a+ e- ae (1) ea = e+ a- e a (2) using ae = a a+ e- ae = a e-ae = 0 e(1-a) = 0 either e = 0 or a = 1 as operation is on Q excluding 1 so a≠1, hence e = 0. So identity element e = 0. (ii) for a binary operation* * if e is identity element then it is invertible with respect to * if for an element b, ab = e = ba where b is called inverse of * and denoted by a^-1. a*b = 0 a+ b- ab = 0 b(1-a) = -a \(b=\frac{-a}{(1-a)}\Rightarrow\frac{a}{(a-1)}\) \(a^{-1}=\frac{a}{(a-1)}\)

Consider a binary operation on Q – {1}, defined by a * b = a + b - ab. (i) Find the identity element in Q – {1}. (ii) Show that each a ∈ Q - {1} has its inverse.

Answer»

(i) For a binary operation *, e identity element exists if a*e = e*a = a. As a*b = a+ b- ab

a*e = a+ e- ae (1)

e*a = e+ a- e a (2)

using a*e = a

a+ e- ae = a

e-ae = 0

e(1-a) = 0

either e = 0 or a = 1 as operation is on Q excluding 1 so a≠1, hence e = 0.

So identity element e = 0.

(ii) for a binary operation * if e is identity element then it is invertible with respect to * if for an element b, a*b = e = b*a where b is called inverse of * and denoted by a^-1.

a*b = 0

a+ b- ab = 0

b(1-a) = -a

\(b=\frac{-a}{(1-a)}\Rightarrow\frac{a}{(a-1)}\)

\(a^{-1}=\frac{a}{(a-1)}\)

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