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Check commutativity and associativity
Given:
\( a * b = a + b – ab, \quad a,b \in \mathbb{Z} \)
Commutativity:
\( a * b = a + b – ab \)
\( b * a = b + a – ba = a + b – ab \)
✔ Operation is commutative
Associativity:
LHS:
\[
(a*b)*c = (a + b – ab)*c
\]
\[
= (a + b – ab) + c – (a + b – ab)c
\]
\[
= a + b + c – ab – ac – bc + abc
\]
RHS:
\[
a*(b*c) = a*(b + c – bc)
\]
\[
= a + (b + c – bc) – a(b + c – bc)
\]
\[
= a + b + c – bc – ab – ac + abc
\]
Both sides are equal:
\( a + b + c – ab – ac – bc + abc \)
✔ Operation is associative
Conclusion:
✔ The operation is both commutative and associative on \( \mathbb{Z} \).