A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is
(a) commutative
(b) associative
(c) not commutative
(d) commutative and associative

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Solution

(c) not commutative

Commutativity:
$Let a, b \in Z a * b = 3 a + b b * a = 3 b + a Thus, a * b \neq b * a If a = 1 and b = 2, 1 * 2 = 3 (1) + 2 = 5 2 * 1 = 3 (2) + 1 = 7 1 * 2 \neq 2 * 1$

Thus, * is not commutative on Z.

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