Q.On $\mathbb{Z}$, define $*$ by $(m*n) = m^n + n^m,\ \forall\, m,n \in \mathbb{Z}$. Is $*$ binary on $\mathbb{Z}$?

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.

Binary operations form part of the Relations and Functions unit in the CBSE Class 12 Mathematics NCERT syllabus, commonly appearing in board exams and searched as "binary operations class 12 important questions" or "commutative associative identity inverse examples". This topic also recurs in JEE Main algebra questions that test identity and inverse-element reasoning.