Question

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

Answer 1

Let a, b ∈ Z

a * b = 3a + 7b

b * a = 3b + 7a

Now, a * b ≠ b * a

Let a = 1 and b = 2

1 * 2 = 3 × 1 + 7 × 2

= 3 + 14

= 17

2 * 1 = 3 × 2 + 7 × 1

= 6 + 7

= 13

So, there exist a = 1, b = 2 ∈ Z such that a * b ≠ b * a

Hence, * is not commutative on Z.