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

Decide whether or not a given binary operator is associative

Determine which of the following binary operations are associative.

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 associative since $(1 \star 1) \star 1 = (1-1)-1 = 0-1 = -1$ but $1 \star (1 \star 1) = 1-(1-1) = 1-0 = 1$ .
Associative since for all $a,b,c$ ,

$(a \star b) \star c$ = $(a + b + ab) \star c$

= $(a + b + ab) + c + ac + bc + abc$

= $a + (b + c + bc) + ab + ac + abc$

= $a \star (b + c+ bc)$

= $a \star (b \star c)$
Not associative since $(1 \star 0) \star 2 = \frac{1}{5} \star 2 = \frac{11}{25}$ but $1 \star (0 \star 2) = 1 \star \frac{2}{5} = \frac{7}{25}$ .
Associative since for all $(a_1,b_1), (a_2,b_2), (a_3,b_3)$ , we have

$(a_1,b_1) \star \left[(a_2, b_2) \star (a_3,b_3) \right]$ = $(a_1,b_1 \star (a_2 b_3 + b_2 a_3, b_2 b_3)$

= $(a_1 b_2 b_3 + b_1 a_2 b_3 + b_1 b_2 a_3, b_1 b_2 b_3)$

= $(a_1 b_2 + b_1 a_2, b_1 b_2) \star (a_3,b_3)$

= $\left[ (a_1,b_1) \star (a_2,b_2) \right] \star (a_3,b_3)$
Not associative since $(2 \star 1) \star 2 = 2 \star 2 = 1$ but $2 \star (1 \star 2) = 2 \star \frac{1}{2} = 4$ .

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