Let * be a binary operation defined on Q. Find which of the following

1.	Let * be a binary operation defined on Q. Find which of the following binary operations are associative(i) a * b = a – b for a, b ∈ Q.(ii) a * b = ab/4 for a, b ∈ Q.(iii) a * b = a – b + ab for a, b ∈ Q.(iv) a * b = ab2 for a, b ∈ Q.
Answer» (i) * is not associative for if we take a = 1, b = 2 and c = 3, then (a * b) * c = (1 * 2) * 3 = (1 – 2) * 3 = – 1 – 3 = – 4 and a * (b * c) = 1 * (2 * 3) = 1 * (2 – 3) = 1 – ( – 1) = 2. Thus (a * b) * c ≠ a * (b * c) and hence * is not associative. (ii) * is associative since Q is associative with respect to multiplication. (iii) * is not associative for if we take a = 2, b = 3 and c = 4, then (a * b) * c = (2 * 3) * 4 = (2 – 3 + 6) * 4 = 5 * 4 = 5 – 4 + 20 = 21, and a * (b * c) = 2 * (3 * 4) = 2 * (3 – 4 + 12) = 2 * 11 = 2 – 11 + 22 = 13 Thus (a * b) * c ≠ a * (b * c) and hence * is not associative. (iv) * is not associative for if we take a = 1, b = 2 and c = 3, then (a * b) * c = (1 * 2) * 3 = 4 * 3 = 4 × 9 = 36 and a * (b * c) = 1 * (2 * 3) = 1 * 18 = 1 × 182 = 324. Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

Let * be a binary operation defined on Q. Find which of the following binary operations are associative(i) a * b = a – b for a, b ∈ Q.(ii) a * b = ab/4 for a, b ∈ Q.(iii) a * b = a – b + ab for a, b ∈ Q.(iv) a * b = ab2 for a, b ∈ Q.

Answer»

(i) * is not associative for if we take a = 1, b = 2 and c = 3, then

(a * b) * c = (1 * 2) * 3 = (1 – 2) * 3 = – 1 – 3 = – 4 and

a * (b * c) = 1 * (2 * 3) = 1 * (2 – 3) = 1 – ( – 1) = 2.

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

(ii) * is associative since Q is associative with respect to multiplication.

(iii) * is not associative for if we take a = 2, b = 3 and c = 4, then

(a * b) * c = (2 * 3) * 4 = (2 – 3 + 6) * 4 = 5 * 4 = 5 – 4 + 20 = 21, and

a * (b * c) = 2 * (3 * 4) = 2 * (3 – 4 + 12) = 2 * 11 = 2 – 11 + 22 = 13

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

(iv) * is not associative for if we take a = 1, b = 2 and c = 3, then (a * b) * c =

(1 * 2) * 3 = 4 * 3 = 4 × 9 = 36 and a * (b * c) = 1 * (2 * 3) = 1 * 18 =

1 × 182 = 324.

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.