NCERT Solutions
Class 12 Maths
Relations and Functions
Ex. 1.4 Q2
For each binary operation * defined below, determine whether * is commutative or associative.
(i) On Z, define a * b = a – b (ii) On Q, define a * b = ab + 1
(iii) On Q, define a * b = ab/2 (iv) On Z+, define a * b = 2ab
(v) On Z+, define a * b = ab (vi) On R − {−1}, define a ∗ b = a/(b + 1)
(i) On Z, * is defined by a * b = a − b
It can be observed that 1 * 2 = 1 − 2 = −1 and 2 * 1 = 2 − 1 = 1
So, 1 * 2 ≠ 2 * 1, where 1, 2 ∈ Z
Hence, the operation * is not commutative.
Also, we have
(1 * 2) * 3 = (1 − 2) * 3 = −1 * 3 = −1 − 3 = −4
1 * (2 * 3) = 1 * (2 − 3) = 1 * −1 = 1 − (−1) = 2
So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Z
Hence, the operation * is not associative.
(ii) On Q, * is defined by a * b = ab + 1
It is known that: ab = ba for all a, b ∈ Q
⇒ ab + 1 = ba + 1 for all a, b ∈ Q
⇒ a * b = a * b for all a, b ∈ Q
Therefore, the operation * is commutative.
It can be observed that
(1 * 2) * 3 = (1 × 2 + 1) * 3 = 3 * 3 = 3 × 3 + 1 = 10
1 * (2 * 3) = 1 * (2 × 3 + 1) = 1 * 7 = 1 × 7 + 1 = 8
So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q
Therefore, the operation * is not associative.
(iii) On Q, * is defined by a * b = ab/2
It is known that: ab = ba for all a, b ∈ Q
⇒ ab/2 = ba/2 for all a, b ∈ Q
⇒ a * b = b * a for all a, b ∈ Q
Therefore, the operation * is commutative.
For all a, b, c ∈ Q, we have
(a ∗ b) ∗ c = (ab/2) ∗ c = {(ab/2) c}/2 = abc/4 and
a ∗ (b ∗ c) = a ∗ (bc/2) = {a(bc/2)}/2 = abc/4
So, (a*b)*c = a*(b*c), where a, b, c ∈ Q
Therefore, the operation * is associative.
(iv) On Z+, * is defined by a * b = 2ab
It is known that: ab = ba for all a, b ∈ Z
⇒ 2ab = 2ba for all a, b ∈ Z+
⇒ a * b = b * a for all a, b ∈ Z+
Therefore, the operation * is commutative.
It can be observed that
(1 ∗ 2) ∗ 3 = 21*2 ∗ 3 = 4 ∗ 3 = 24*3 = 212 and
1 ∗ (2 ∗ 3) = 1 ∗ 22*3 = 1 ∗ 26 = 1 ∗ 64 = 21*64 = 264
So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Z+
Therefore, the operation * is not associative.
(v) On Z+, * is defined by a * b = ab
It can be observed that
1 * 2 = 12 = 1 and 2*1 = 21 = 2
So, 1 * 2 ≠ 2 * 1, where 1, 2 ∈ Z+
Therefore, the operation * is not commutative.
It can also be observed that
(2 ∗ 3) ∗ 4 = 23 ∗ 4 = 8 ∗ 4 = 84 = 212 and
2 ∗ (3 ∗ 4) = 2 ∗ 34 = 2 ∗ 81 = 281
So, (2 * 3) * 4 ≠ 2 * (3 * 4), where 2, 3, 4 ∈ Z+
Therefore, the operation * is not associative.
(vi) On R, * − {−1} is defined by a ∗ b = a/(b + 1)
It can be observed that
1 ∗ 2 = 1/(2 + 1) = 1/3 and 2 ∗ 1 = 2/(1 + 1) = 2/2 = 1
So, 1 * 2 ≠ 2 * 1, where 1, 2 ∈ R − {−1}
Therefore, the operation * is not commutative.
It can also be observed that
(1 ∗ 2) ∗ 3 = 1/(2 + 1) * 3 = 1/3 * 3 = (1/3)/(3 + 1) = 1/12 and
1 ∗ (2 ∗ 3) = 1 ∗ 2/(3 + 1) = = 1 ∗ (2/4) = 1 ∗ 1/2 = 1/(1/2 + 1) = 1/(3/2) = 2/3
So, (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ R − {−1}
Therefore, the operation * is not associative.