A binary operation * on Z defined by a * b = 3a + b for all a, b … - Vidyadip

Question

A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is:

Commutative.
Associative.
Not commutative.
Commutative and associative.

✓

Answer

Not commutative.

Solution:
Let $\text{a, b}\in\text{Z}$
a * b = 3a + b
b * a = 3b + 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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