Algebra Solutions: Decide whether or not a given binary operator is commutative

Decide whether or not a given binary operator is commutative

Determine which of the following binary operations are commutative.

The operation $\star$ on $\mathbb{Z}$ defined by $a \star b = a-b$ .
The operation $\star$ on $\mathbb{R}$ defined by $a \star b = a+b+ab$ .
The operation $\star$ on $\mathbb{Q}$ defined by $a \star b = \frac{a+b}{5}$ .
The operation $\star$ on $\mathbb{Z} \times \mathbb{Z}$ defined by
$(a_1,b_1) \star (a_2,b_2) = (a_1 b_2 + b_1 a_2, b_1 b_2)$ .
The operation $\star$ on $\mathbb{Q} \setminus \{0\}$ defined by $a \star b = \frac{a}{b}$ .

Not commutative since $1 \star (-1) = 1 - (-1) = 2$ but $(-1) \star 1 = -1 - 1 = -2$ .
Commutative since $a \star b = a + b + ab = b + a + ba = b \star a$ .
Commutative since $a \star b = \frac{a+b}{5} = \frac{b+a}{5} = b \star a$ .
Commutative since

$(a_1,b_1) \star (a_2,b_2)$ = $(a_1 b_2 + b_1 a_2, b_1 b_2)$

= $(a_2 b_1 + b_2 a_1, b_2 b_1)$

= $(a_2,b_2) \star (a_1,b_1)$ .
Not commutative since $1 \star 2 = \frac{1}{2}$ but $2 \star 1 = 2$ .

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