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Let Q0 be the set of all nonzero rational numbers. Let * be a binary operation on Q0, defined by a * b = ab/4 for all a, b ∈ Q0.

(i) Show that * is commutative and associative.

(ii) Find the identity element in Qo.

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

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It is given that a * b = ab/4

(i) For a, b, c ∈ Q0

We know that

a * b = ab/4 = ba/4 = b * a

(a * b) * c = ab/ 4 * c = [ab/4 * c]/ 4 = (ab) c/ 16

a * (b * c) = a * bc/4 = [a(bc/4)]/ 4 = a (bc)/16

Here, (ab) c = a (bc)

Therefore, (a * b) * c = a * (b * c)

(ii) Consider e as the identity element and a ∈ Q0

Here, a * e = a

So we get

ae/4 = a where e = 4

Hence, 4 is the identity element in Q.

(iii) Consider a ∈ Q0 which is inverse b

a * b = e

So we get

ab/4 = 4

Here, b = 16/a ∈ Q0

Hence, a ∈ Q0 has 16/a as inverse.

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