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Question
On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , b ∈ Z. Prove that * is not associative on Z.
Solution
\[\text{Let }a, b, c \in Z\]
\[a * \left( b * c \right) = a * \left( bc + 1 \right)\]
\[ = a\left( bc + 1 \right) + 1\]
\[ = abc + a + 1\]
\[\left( a * b \right) * c = \left( ab + 1 \right) * c\]
\[ = \left( ab + 1 \right)c + 1\]
\[ = abc + c + 1\]
\[\text{Thus,a} * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Z.
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