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Question
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
Options
commutative
associative
not commutative
commutative and associative
Solution
not commutative
Commutativity:
\[\text{ Let } a, b \in Z\]
\[a * b = 3a + b\]
\[b * a = 3b + a\]
\[\text{ Thus },a * b \neq b * a\]
\[\text{ If a = 1 and b } = 2, \]
\[1 * 2 = 3\left( 1 \right) + 2\]
\[ = 5\]
\[2 * 1 = 3\left( 2 \right) + 1\]
\[ = 7\]
\[1 * 2 \neq 2 * 1\]
Thus, * is not commutative on Z .
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