Q.Let $A=\{a+\sqrt5\,b : a,b\in\mathbb{Z}\}$. Check whether the usual multiplication is a binary operation on $A$.

Binary Operations

A binary operation $*$ on a set $S$ assigns to each ordered pair $(a,b)$ an element $a*b\in S$ (closure). Its algebraic properties are:

• Commutative: $a*b=b*a$ for all $a,b$.
• Associative: $(a*b)*c=a*(b*c)$ for all $a,b,c$.
• Identity element $e$: $a*e=e*a=a$ for all $a$.
• Inverse of $a$: an element $a'$ with $a*a'=a'*a=e$.

To analyse a given rule such as $a*b=\dfrac{a+b}{2}$ or $a*b=a+b+ab$, check each property directly. For $a*b=a+b+ab$: it is commutative and associative, the identity solves $a+e+ae=a\Rightarrow e=0$, and the inverse solves $a+a'+aa'=0\Rightarrow a'=-\dfrac{a}{1+a}$.

Exam questions define an operation on $\mathbb R$ (or a subset) and ask whether it is commutative/associative, or to find its identity and inverse elements. …