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Question
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
Solution
\[\text{Let a}, b \in Z . {Then}, \]
\[a * b = 3a + 7b\]
\[b * a = 3b + 7a\]
\[\text{Thus, a} * b \neq b * a\]
\[\text{Let a} = 1 \text{ and } b = 2 \]
\[1 * 2 = 3 \times 1 + 7 \times 2\]
\[ = 3 + 14\]
\[ = 17\]
\[2 * 1 = 3 \times 2 + 7 \times 1\]
\[ = 6 + 7\]
\[ = 13\]
\[\text{Therefore}, \exists \text{ a} = 1; b = 2 \in \text{Z such that} a * b \neq b * a\]
Thus, * is not commutative on Z.
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