Q.(i) Define an operation $*$ on $\mathbb{Q}$ as follows: $a*b=\left(\dfrac{a+b}{2}\right);\ a,b\in\mathbb{Q}$. Examine the closure, commutative, and associative properties satisfied by $*$ on $\mathbb{Q}$.

Once an operation $*$ is confirmed binary on a set $S$ (so closure automatically holds), four more properties decide how "arithmetic-like" it behaves.

Commutative property. $*$ is commutative if $a*b=b*a$ for every $a,b\in S$ -- order does not matter. Ordinary $+,\times$ on numbers are commutative; ordinary $-$ is not ($4-5\ne5-4$).

Associative property. $*$ is associative if $(a*b)*c=a*(b*c)$ for every $a,b,c\in S$ -- grouping does not matter, so a chain $a*b*c$ is unambiguous. Ordinary $-$ fails this too: $(4-5)-7=-8$ but $4-(5-7)=6$.

Existence of identity. An element $e\in S$ is an identity element for $*$ if $a*e=a=e*a$ for every $a\in S$. For $+$ on $\mathbb Z$, $e=0$; for $\times$ on $\mathbb Q$, $e=1$.

Existence of inverse. If an identity $e$ exists, then $b\in S$ is the inverse of $a$ (written $b=a^{-1}$) if $a*b=e=b*a$. For $+$ on $\mathbb Z$, the inverse of $m$ is $-m$; for $\times$ on $\mathbb Q$, the inverse of a nonzero $x$ is $\tfrac1x$. (The notation $a^{-1}$ names an element, not the fraction $\tfrac1a$.)

Uniqueness is guaranteed, not assumed.

Theorem 12.1 (Uniqueness of Identity). If an algebraic structure $(S,*)$ has an identity element, it has only one. Proof idea: if $e_1,e_2$ are both identities, treat $e_1$ as the identity acting on $e_2$ to get $e_1*e_2=e_2$, then treat $e_2$ as the identity acting on $e_1$ to get $e_1*e_2=e_1$; comparing gives $e_1=e_2$.

Theorem 12.2 (Uniqueness of Inverse). If $a\in S$ has an inverse, it has only one. Proof idea: if $a_1,a_2$ are both inverses of $a$, then $a_1=a_1*e=a_1*(a*a_2)=(a_1*a)*a_2=e*a_2=a_2$. …