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

Answer 1

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 .