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Question
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?
Solution
Commutativity:
\[\text{Let }a, b \in Z . \text{Then}, \]
\[a * b = a + b - 4\]
\[ = b + a - 4\]
\[ = b * a\]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in Z\]
Thus, * is commutative on Z.
Associativity:
\[\text{Let }a, b, c \in Z . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b + c - 4 \right)\]
\[ = a + b + c - 4 - 4\]
\[ = a + b + c - 8\]
\[\left( a * b \right) * c = \left( a + b - 4 \right) * c\]
\[ = a + b - 4 + c - 4\]
\[ = a + b + c - 8\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Z\]
Thus, * is associative on Z.
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