Define * on Z by a * b = a + b - ab. Show that * is a binary operation on Z which is commutative as well as associative.

* is an operation as a*b = a+ b - ab where a, b ∈ Z. Let and b = 2 two integers.

So, * is a binary operation from .

For commutative,

Since a*b = b*a, hence * is a commutative binary operation.

Again for associative,

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

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

= a+ b+ c- bc- ab- ac+ abc

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

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

= a+ b+ c- ab- ac- bc+ abc

As a*(b*c) = (a*b) *c, hence * an associative binary operation.