Def 1
Let $S$ be a set. A binary operation on $S$ is a fuction $* : S \times S \to S$
# $(a, b) \mapsto *(a, b)$
Notation
$*(a, b) = a*b \; (=ab)$
Def 2
A binary operation $*$ on a set $S$ is associative if $(a * b) * c = a * (b * c)$ for all $a, b, c \in S$.
(We can write $a * b * c$)
Def 3
A binary operation $*$ on a set $S$ is commutative if $a * b = b * a$ for all $a, b \in S$.
Example
| set | operation | binary | associative | commutative |
| $\mathbb{Z}$ | $a * b = a + b$ | O | O | O |
| $\mathbb{R}$ | $a * b = a \times b$ | O | O | O |
| $\mathbb{R} \setminus \left\{0 \right\}$ | $a * b = a \div b$ | O | X | X |
| $M_{n}(\mathbb{R})$ | $A * B = AB$ | O | O | X |
| $\mathbb{Z}$ | $n \mapsto 2n$ $(\mathbb{Z} \to \mathbb{Z})$ |
X | ||
| $V$ (scalar multiplication) |
$(r, \mathbf{u}) \mapsto r \mathbf{u}$ $(\mathbb{R} \times V \to V)$ |
X | ||
| $V$ (inner product) |
$(\mathbf{u}, \mathbf{v}) \mapsto \mathbf{u} \cdot \mathbf{v}$ $(V \times V \to \mathbb{R})$ |
X | ||
| $M_{n}(\mathbb{R})$ | $A*B = AB - BA$ | O | X | X |
| $\mathbb{R}^{3}$ (cross product) |
$\mathbf{u}*\mathbf{v} = \mathbf{u} \times \mathbf{v}$ | O | X | X |
Def 4
$S$ : set
$*$ : binary operation on $S$
Let $H$ be a subset of $S$.
1. $H$ is closed under $*$ if $a * b \in H$ for all $a, b \in H$.
2. In this case, $* : H \times H \to H$ is called the induced operator of $*$ on $H$.
Ex
$\div$ on $\mathbb{R}^{+}$ : $\mathbb{Z}^{+} \subset \mathbb{R}^{+}$
$\Rightarrow$ $\mathbb{Z}^{+}$ is not closed under $\div$
$+, \times$ on $\mathbb{Z}$ : $\vartheta \overset{\underset{\mathrm{def}}{}}{=} \left\{2n+1 \; | n \in \mathbb{Z} \right\} \subset \mathbb{Z}$
$\Rightarrow$ $\vartheta$ is not closed under $+$, but $\vartheta$ is closed under $\times$