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प्रश्न
Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all a, b ∈ Z ?
उत्तर
Commutativity:
\[\text{Let a, b} \in Z . \text{Then}, \]
\[a * b = a + b + ab\]
\[ = b + a + ba\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in Z\]
Associativity :
\[\text{Let } a, b, c \in Z . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\]
\[ = a + \left( b + c + bc \right) + a\left( b + c + bc \right)\]
\[ = a + b + c + bc + ab + ac + abc\]
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\]
\[ = a + b + ab + c + \left( a + b + ab \right)c\]
\[ = a + b + ab + c + ac + bc + abc\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Z\]
Thus, * is associative on Z.
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