Question

Define $\star$ on Z by $a\star b=a-b+ab$.
Show that $\star$ is a binary operation on Z which is neither commutative nor associative.

Answer 1

A binary operation on a set is an operation whose domains and the codomain are the same set.

$a\in Z,b\in Z\Rightarrow (a-b)\in Z$ and $ab\in Z$
$\Rightarrow \left\{\right(a,b)+ab\}\in Z\Rightarrow a-b+ab\in Z$.
$\therefore Z$ is closed for $\star$.
Show that $3\star 2\ne 2\star 3$ and $(4\star 3)\star 2\ne 4\star (3\star 2)$.