Show that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_*$ of nonzero real numbers.

(N/A) binary operation $*$ on a set $S$ is a function $*: S \times S \rightarrow S$.
$1$. For addition $(+)$: For any $a, b \in R$, $a+b$ is a unique real number. Thus, $+: R \times R \rightarrow R$ is a binary operation.
$2$. For subtraction $(-)$: For any $a, b \in R$, $a-b$ is a unique real number. Thus, $-: R \times R \rightarrow R$ is a binary operation.
$3$. For multiplication $(\times)$: For any $a, b \in R$, $a \times b$ is a unique real number. Thus, $\times: R \times R \rightarrow R$ is a binary operation.
$4$. For division $(div)$: For $a, b \in R$, the operation $a \div b = \frac{a}{b}$ is not defined when $b=0$. Since $0 \in R$, the division operation is not a function from $R \times R$ to $R$. Hence, it is not a binary operation on $R$.
$5$. For $R_*$ (the set of nonzero real numbers): For any $a, b \in R_*$, $a \neq 0$ and $b \neq 0$. The quotient $\frac{a}{b}$ is always a defined real number, and since $a, b \neq 0$, $\frac{a}{b} \neq 0$. Thus, $\frac{a}{b} \in R_*$. Therefore, division is a binary operation on $R_*$.