Let Q0 be the set of all nonzero rational numbers. Let * be a binary

1.	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.
Answer» 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.

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.

Answer»

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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