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 $R_*$.
$1$. 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 $R_*$.
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Consider a binary operation $*$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^* \,3) \,^* \,4$ and $2 \,^* \,(3 \,^* \,4)$.
$^*$ $1$ $2$ $3$ $4$ $5$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $1$ $2$ $2$ $2$ $2$ $3$ $1$ $2$ $3$ $3$ $3$ $4$ $1$ $2$ $3$ $4$ $4$ $5$ $1$ $2$ $3$ $4$ $5$
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| $2$ | $1$ | $2$ | $2$ | $2$ | $2$ |
| $3$ | $1$ | $2$ | $3$ | $3$ | $3$ |
| $4$ | $1$ | $2$ | $3$ | $4$ | $4$ |
| $5$ | $1$ | $2$ | $3$ | $4$ | $5$ |
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