Let Q 0 be the set of all nonzero rational numbers. Let * be a binary operation on Q 0 , defined by for all a, b ∈ Q 0 . (i) Show that * is commutative and associative. (ii) Find the identity element in Q 0 . (iii) Find the inverse of an element a in Q 0 .

Question

Philoid · Accepted Answer

(i) For commutative binary operation, a*b = b*a.

as multiplication is commutative ab = ba so a*b = b*a. Hence * is commutative binary operation.

For associative binary operation, a*(b*c) = (a*b) *c

Since a*(b*c) = (a*b) *c, hence * is an associative binary operation.

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

(1)

(2)

using a*e = a

Either a = 0 or e = 4 as given a≠0, so e = 4.

Identity element e = 4.

(iii) 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 = 4

Let Q0 be the set of all nonzero rational numbers. Let * be a binary operation on Q0, defined by for all a, b ∈ Q0.(i) Show that * is commutative and associative.(ii) Find the identity element in Q0.(iii) Find the inverse of an element a in Q0.

Let Q₀ be the set of all nonzero rational numbers. Let * be a binary operation on Q₀, defined by for all a, b ∈ Q₀.
(i) Show that * is commutative and associative.

(ii) Find the identity element in Q₀.

(iii) Find the inverse of an element a in Q₀.