Question

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

Answer 1

$$\begin{gathered} \text{Let}a,b\in Z.\mathrm{Then}, \\ a*b=3a+7b \\ b*a=3b+7a \\ \text{Thus,}a*b\ne b*a \\ \text{Let}a=1\mathrm{and}b=2 \\ 1*2=3\times 1+7\times 2 \\ =3+14 \\ =17 \\ 2*1=3\times 2+7\times 1 \\ =6+7 \\ =13 \\ \text{Therefore, ∃}a=1;b=2\in Z\text{such that}a*b\ne b*a \end{gathered}$$

Thus, * is not commutative on Z.