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## Decide whether or not a given binary operator is commutative

Determine which of the following binary operations are commutative.

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

Solution:

(1) Not commutative since $$1 \star (-1) = 1 - (-1) = 2$$ but $$(-1) \star 1 = -1 - 1 = -2.$$

(2) Commutative since $$a \star b = a + b + ab = b + a + ba = b \star a.$$

(3) Commutative since $$a \star b = \frac{a+b}{5} = \frac{b+a}{5} = b \star a.$$

(4) Commutative since \begin{align*} (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).\end{align*}

(5) Not commutative since $1 \star 2 = \frac{1}{2}$ but $2 \star 1 = 2$.