Let * be the binary operation on Q of rational numbers as follows:
i. a * b = a - b
ii. a * b = a² + b²
iii. a * b = a + ab
iv. a * b = (a - b)²
v. a + b = ab/4
vi. a * b = ab²
Find which of the binary operations are commutative and which are associative

Solution:

The commutative property deals with the arithmetic operations of addition and multiplication. It means that changing the order or position of numbers while adding or multiplying them does not change the end result

(i). On Q, the operation * is defined as a * b = a - b

1/2 * 1/3 = 1/2 - 1/3 = (3 - 2)/6 = 1/6

And

1/3 * 1/2 = 1/3 - 1/2 = (2 - 3)/6 = - 1/6

(1/2 * 1/3) ≠ (1/3 * 1/2) where 1/2 , 1/3 ∈ Q

⇒Operation * is not commutative.

1/2 * (1/3 * 1/4) = 1/2 * (1/3 - 1/4) = 1/2 * 1/12 = 1/2 - 1/12 = (6 - 1)/12 = 5/12

(1/2 * 1/3) * 1/4 ≠ 1/2 * (1/3 * 1/4) where 1/2 , 1/3, 1/4 ∈ Q

⇒ Operation * is not associative.

(ii). On Q, the operation * is defined as a * b = a² + b²

For a, b ∈ Q

a * b = a² + b² = b² + a² = b * a

⇒ a * b = b * a

⇒ Operation * is commutative.

(1 * 2) * 3 = (1² + 2²)* 3 = (1 + 4) * 3 = 5 * 3 = 5² + 3² = 25 + 9 = 34

1 * (2 * 3) = 1 * (2² + 3²) = 1 * (4 + 9) = 1 * 13 = 1² + 13² = 1 + 169 = 170

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3) where 1, 2, 3 ∈ Q

⇒ Operation * is not associative.

iii. On Q, the operation * is defined as a * b = a + ab

1 * 2 = 1 + 1 × 2 = 1 + 2 = 3

2 * 1 = 2 + 2 × 1 = 2 + 2 = 4

⇒ 1 * 2 ≠ 2 * 1 where 1, 2 ∈ Q

⇒ Operation * is not commutative.

(1 * 2) * 3 = (1 + 1 × 2) * 3 = 3 * 3 = 3 + 3 × 3 = 3 + 9 = 12

1 * (2 * 3) = 1 * (2 + 2 × 3) = 1 * 8 = 1 + 1 × 8 = 1 + 8 = 9

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3) where1, 2, 3 ∈ Q

Operation * is not associative.

iv. On Q, the operation * is defined as a * b = (a - b)²

For a, b ∈ Q

a * b = (a - b)²

b * a = (b - a)² = [- (a - b)]² = (a - b)²

⇒ a * b = b * a

⇒ Operation * is commutative.

(1 * 2) * 3 = (1 - 2)² * 3 = (- 1)² * 3 = 1 * 3 = (1 - 3)² = (- 2)² = 4

1 * (2 * 3) = 1 * (2 - 3)² = 1 * (- 1)² = 1 * 1 = (1 - 1)² = 0

⇒ (1 * 2) * 3 ≠ 1 * (2 * 3) where 1, 2, 3 ∈ Q

⇒ Operation * is not associative.

v. On Q, the operation * is defined as a + b = ab/4

For a, b ∈ Q

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

⇒ a * b = b * a

⇒ Operation * is commutative.

For a, b, c ∈ Q

(a * b) * c = ab/4 * c = (ab/4 . c)/4 = abc/16

a * (b * c) = a * ab//4 = (c. ab/4)/4 = abc/16

\(a * b) * c = a * (b * c) where a, b, c ∈ Q

⇒ Operation * is associative.

vi. On Q, the operation * is defined as a * b = ab²

1/2 * 1/3 = 1/2 * (1/3)² = 1/2. 1/9 = 1/18

1/3 * 1/2 = 1/3 * (1/2)² = 1/3. 1/4 = 1/12

⇒ (1/2 * 1/3) ≠ (1/3 * 1/2) where 1/2, 1/3 ∈ Q

⇒ Operation * is not commutative.

(1/2 * 1/3) * 1/4 = (1/2. (1/3)²) * 1/4 = 1/18 * 1/4 = 1/18. (1/4)² = 1/(18 × 16)

1/2 * (1/3 * 1/4) = 1/2 * (1/3 * (1/4)²) = 1/2 * 1/48 = 1/2. (1/48)² = 1/(2. (48)²)

(1/2 * 1/3) * 1/4 ≠ 1/2 * (1/3 * 1/4) where 1/2, 1/3, 1/4 ∈ Q

⇒ Operation * is not associative.

Operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.4 Question 9

Let * be the binary operation on Q of rational numbers as follows: (i). a * b = a - b (ii). a * b = a² + b² (iii). a * b = a + ab (iv). a * b = (a - b)² (v). a + b = ab/4 (vi). a * b = ab² . Find which of the binary operations are commutative and which are associative

Summary:

For the given binary operation on Q of rational numbers. Operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative