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Prove that the operation is not associative
Given:
\( a * b = \frac{a-b}{2}, \quad a,b \in \mathbb{Q} \)
Proof (Counterexample Method):
Take \( a = 2 \), \( b = 4 \), \( c = 6 \)
LHS:
\( (a*b)*c = \left(\frac{2-4}{2}\right)*6 = (-1)*6 \)
\( = \frac{-1 – 6}{2} = \frac{-7}{2} \)
RHS:
\( a*(b*c) = 2*\left(\frac{4-6}{2}\right) = 2*(-1) \)
\( = \frac{2 – (-1)}{2} = \frac{3}{2} \)
Clearly:
\( (a*b)*c \neq a*(b*c) \)
Conclusion:
❌ Therefore, the operation is NOT associative on \( \mathbb{Q} \).