Mark the tick against the correct answer in the following: Let Q+ be

1.	Mark the tick against the correct answer in the following: Let Q+ be the set of all positive rationals. Then, the operation * on Q+ defined by \(a*b=\frac{ab}{2}\) for all a, b ∈ Q+ isA. commutative but not associative B. associative but not commutative C. neither commutative nor associative D. both commutative and associative
Answer» Correct Answer is (D) both commutative and associative According to the question , Q = { Positive rationals } R = {(a, b) : a * b = ab/2 } Formula * is commutative if a * b = b * a * is associative if (a * b) * c = a * (b * c) Check for commutative Consider , a * b = ab/2 And , b * a = ba/2 Both equations are the same and will always true . Therefore , * is commutative ……. (1) Check for associative Consider , (a * b) * c = (ab/2) * c = \(\frac{\frac{ab}{2}\times c}{2}\)= abc/4 And , a * (b * c) = a * (bc/2) = \(\frac{a\times\frac{bc}{2}}{2}\)= abc/4 Both the equation are the same and therefore will always be true. Therefore , * is associative ……. (2) Now , according to the equations (1) , (2) Correct option will be (D)

Mark the tick against the correct answer in the following: Let Q+ be the set of all positive rationals. Then, the operation * on Q+ defined by \(a*b=\frac{ab}{2}\) for all a, b ∈ Q+ isA. commutative but not associative B. associative but not commutative C. neither commutative nor associative D. both commutative and associative

Answer»

Correct Answer is (D) both commutative and associative

According to the question ,

Q = { Positive rationals }

R = {(a, b) : a * b = ab/2 }

Formula

* is commutative if a * b = b * a

* is associative if (a * b) * c = a * (b * c)

Check for commutative

Consider , a * b = ab/2

And , b * a = ba/2

Both equations are the same and will always true .

Therefore , * is commutative ……. (1)

Check for associative

Consider , (a * b) * c = (ab/2) * c = \(\frac{\frac{ab}{2}\times c}{2}\)= abc/4

And , a * (b * c) = a * (bc/2) = \(\frac{a\times\frac{bc}{2}}{2}\)= abc/4

Both the equation are the same and therefore will always be true.

Therefore , * is associative ……. (2)

Now , according to the equations (1) , (2)

Correct option will be (D)