Question

Let * be the binary operation defined on Q. Which of the following binary operations are commutative?

• A. a * b = a – b ∀ a, b ∈ Q
• B. a * b = $a^2+b^2$ ∀ a, b ∈ Q
• C. a * b = a + ab ∀ a, b ∈ Q
• D. a * b =$(a–b{)}^2$ ∀ a, b ∈ Q

Answer 1

Answer: (B), (D)

The correct options are
B a * b = $a^2+b^2$ ∀ a, b ∈ Q
D a * b =$(a–b{)}^2$ ∀ a, b ∈ Q

(i) a∗b = a − b ∀ a,b ∈ Q
but b∗a = b − a ≠ a − b ∀ a,b ∈ Q
Hence * operation is not commutative

(ii) a∗b = $a^2+b^2$ ∀ a,b ∈ Q
b∗a = $b^2+a^2=a^2+b^2$
[addition is commutaive in Q]
Therefore, a∗b=b∗a
* operation is commutative

(iii) a∗b=a + ab ∀ a,b ∈ Q
But b∗a = b + ba
Clearly a∗b ≠ b∗a
* operation is not commutative

(iv) a∗b=$(a-b{)}^2$ ∀ a,b ∈ Q
b∗a=$(b-a{)}^2={\left(-(a-b)\right]}^2=(a-b{)}^2$
a∗b=b∗a
* operation is commutative