Define * on Z by a * b = a – b + ab. Show that * is a binary operation on Z which is neither commutative nor associative.

Question

Philoid · Accepted Answer

* 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 not commutative 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 * not an associative operation.