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Which of the following statements is/are true: If * is a binary operation on Q where Q is set of rational numbers such that a * b = ab + 1 ∀ a, b ∈ Q. 1. * is commutative on Q. 2. * is associative on Q.

Which of the following statements is/are true: If * is a binary operation on Q where Q is set of rational numbers such that a * b = ab + 1 ∀ a, b ∈ Q. 1. * is commutative on Q. 2. * is associative on Q. Correct Answer Only 1

Concept:

Let * be a binary operation on a non-empty set S. Then

  • * is associative on S if (a * b) * c = a * (b * c) ∀ a, b, c ∈ S.
  • * is commutative on S if a * b = b * a ∀ a, b ∈ S.

Calculation:

Given: * is a binary operation on Q where Q is set of rational numbers such that a * b = ab + 1 ∀ a, b ∈ Q.

Statement 1: * is commutative on Q.

Let a, b ∈ Q

First find out a * b

According to the definition of * we have

⇒ a * b = ab + 1----------(1)

Similarly lets find out b * a

According to the definition of * we have

⇒ b * a = ba + 1-----------(∵ Multiplication is commutative on Q)

⇒ b * a = ab + 1-----------(2)

Now from (1) and (2), we have a * b = b * a ∀ a, b ∈ Q.

Hence, statement 1 is true.

Statement 2: * is associative on Q.

Let a, b, c ∈ Q.

First find out (a * b) * c

According to the definition of * we have

⇒ (a * b) * c = (ab + 1) * c = (ab + 1)c + 1

⇒ (a * b) * c = abc + c + 1---------------(3)

Similarly, lets find out a * (b * c)

According to the definition of * we have

⇒ a * (b * c) = a * (bc + 1) = a (bc + 1) + 1

⇒ a * (b * c) = abc + a + 1-------(4)

Now from (3) and (4) we have (a * b) * c ≠ a * (b * c) ∀ a, b, c ∈ Q

Hence, statement 2 is not true.

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