Question

Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?

Answer 1

\[\text{Let a}, b \in Z . {Then}, \]

\[a * b = 3a + 7b\]

\[b * a = 3b + 7a\]

\[\text{Thus, a} * b \neq b * a\]

\[\text{Let a} = 1 \text{ and } b = 2 \]

\[1 * 2 = 3 \times 1 + 7 \times 2\]

         \[ = 3 + 14\]

          \[ = 17\]

\[2 * 1 = 3 \times 2 + 7 \times 1\]

       \[ = 6 + 7\]

      \[ = 13\]

\[\text{Therefore}, \exists \text{ a} = 1; b = 2 \in \text{Z such that} a * b \neq b * a\]

Thus, * is not commutative on Z.