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

1.	Mark the tick against the correct answer in the following: Let Z be the set of all integers. Then, the operation * on Z defined by a * b = a + b - ab 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 (D) both commutative and associative According to the question , Q = { All integers } R = {(a, b) : a * b = a + b - ab } 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 - ab And , b * a = b + a - ba Both equations are the same and will always be true . Therefore , * is commutative ……. (1) Check for associative Consider , (a * b) * c = (a + b - ab) * c = a + b - ab + c -(a + b - ab)c =a + b – ab + c – ac – bc + abc And , a * (b * c) = a * (b + c - bc) = a + (b + c - bc) - a(b + c - bc) =a + b + c – bc - ab – ac + abc 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 Z be the set of all integers. Then, the operation * on Z defined by a * b = a + b - ab 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 (D) both commutative and associative

According to the question ,

Q = { All integers }

R = {(a, b) : a * b = a + b - ab }

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 - ab

And , b * a = b + a - ba

Both equations are the same and will always be true .

Therefore , * is commutative ……. (1)

Check for associative

Consider , (a * b) * c = (a + b - ab) * c

= a + b - ab + c -(a + b - ab)c

=a + b – ab + c – ac – bc + abc

And , a * (b * c) = a * (b + c - bc)

= a + (b + c - bc) - a(b + c - bc)

=a + b + c – bc - ab – ac + abc

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)