Determine with brief justification which of the following bi

Determine (with brief justification) which of the following binary operations are associative.

(a) the operation on Z defined by a b = a b

(b) the operation on Q defined by a b = a + b 5

(a) let Z be the set of all integers and * be the operation defined on Z such that a*b = a - b

Then for any a , b, c Z ,

consider a * (b*c) = a* (b - c) = a - (b - c) = a - b + c

now consider ( a*b) * c = (a - b)*c = (a - b) - c = a - b - c

Since a * (b * c) ( a * b) * c and hence * is not associative on Z.

(b) let Q - {0} be the set of all rational numbers and * be the operation defined on Q such that a*b = a + b /5

Then for any a , b, c Q - {0} ,

consider a * (b*c) = a* (b + c/5 ) = a + (b + c /5) /5

now consider ( a*b) * c = (a + b/5) + c /5

Since a * (b * c) ( a * b) * c and hence * is not associative on Q.

Then for any a , b, c Q ,

consider a * (b*c) = a* (bc) = abc

now consider ( a*b) * c = (ab)*c = abc

Since a * (b * c) = ( a * b) * c and hence * is associative on Q.