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Show that the operation is not commutative
Given:
\( a * b = 3a + 7b, \quad a,b \in \mathbb{Z} \)
Proof (Counterexample Method):
Take \( a = 1 \), \( b = 2 \)
\( a * b = 3(1) + 7(2) = 3 + 14 = 17 \)
\( b * a = 3(2) + 7(1) = 6 + 7 = 13 \)
Clearly:
\( a * b \neq b * a \)
Conclusion:
❌ Therefore, the operation is NOT commutative on \( \mathbb{Z} \).