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Prove that the operation is not associative
Given:
\( a * b = ab + 1, \quad a,b \in \mathbb{Z} \)
Proof (Counterexample Method):
Take \( a = 1 \), \( b = 2 \), \( c = 3 \)
LHS:
\( (a*b)*c = (1*2)*3 \)
\( 1*2 = (1)(2) + 1 = 3 \)
\( (1*2)*3 = 3*3 = (3)(3) + 1 = 10 \)
RHS:
\( a*(b*c) = 1*(2*3) \)
\( 2*3 = (2)(3) + 1 = 7 \)
\( 1*(2*3) = 1*7 = (1)(7) + 1 = 8 \)
Clearly:
\( (a*b)*c \neq a*(b*c) \)
Conclusion:
❌ Therefore, the operation is NOT associative on \( \mathbb{Z} \).