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

1.	Mark the tick against the correct answer in the following: Let Z + be the set of all positive integers. Then, the operation * on Z + defined bya * b = a b is A. commutative but not associative B. associative but not commutative C. neither commutative nor associative D. both commutative and associative
Answer» Correct Answer is (C) neither commutative nor associative According to the question , Q = { All integers } R = {(a, b) : a * b = a^b } Formula * is commutative if a * b = b * a * is associative if (a * b) * c = a * (b * c) Check for commutative *Consider , a b = a^b** And , b * a = b^a Both equations are not the same and will not always be true . Therefore , * is not commutative ……. (1) Check for associative *Consider , (a b) * c = (a^b) * c =\((a^b)^c\)** And , a * (b * c) = a * (b^c )=\(a^{(b^c)}\) Ex a=2 b=3 c=4 (a * b) * c = (2³) * c =\((8)^4\) a * (b * c) = 2 * (3⁴)=\(2^{(81)}\) Both the equation are not the same and therefore will not always be true. Therefore , * is not associative ……. (2) Now , according to the equations (1) , (2) Correct option will be (C)

Mark the tick against the correct answer in the following: Let Z + be the set of all positive integers. Then, the operation * on Z + defined bya * b = a b is A. commutative but not associative B. associative but not commutative C. neither commutative nor associative D. both commutative and associative

Answer»

Correct Answer is (C) neither commutative nor associative

According to the question ,

Q = { All integers }

R = {(a, b) : a * b = a^b }

Formula

* is commutative if a * b = b * a

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

Check for commutative

Consider , a * b = a^b

And , b * a = b^a

Both equations are not the same and will not always be true .

Therefore , * is not commutative ……. (1)

Check for associative

Consider , (a * b) * c = (a^b) * c =\((a^b)^c\)

And , a * (b * c) = a * (b^c )=\(a^{(b^c)}\)

Ex a=2 b=3 c=4

(a * b) * c = (2³) * c =\((8)^4\)

a * (b * c) = 2 * (3⁴)=\(2^{(81)}\)

Both the equation are not the same and therefore will not always be true.

Therefore , * is not associative ……. (2)

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

Correct option will be (C)