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Question
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?
Solution
Commutativity :
\[\text{Let }a, b \in Z . \text{Then}, \]
\[a * b = a - b\]
\[b * a = b - a\]
\[\text{Therefore},\]
\[a * b \neq b * a\]
Thus, * not is commutative on Z.
Associativity:
\[\text{Let }a, b, c \in Z . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b - c \right)\]
\[ = a - \left( b - c \right)\]
\[ = a - b + c\]
\[\left( a * b \right) * c = \left( a - b \right) - c\]
\[ = a - b - c\]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Z.
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