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Question
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ________________ .
Options
commutative only
associative only
commutative and associative both
none of these
Solution
commutative and associative both
Commutativity :
\[\text{Let }a, b \in N\]
\[\text{ Then }, \]
\[a * b = a + b + ab\]
\[ = b + a + ba\]
\[ = b * a\]
\[\text{Thus},\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity:
\[\text{ Let } a, b, c \in N\]
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\]
\[ = a + b + c + bc + a \left( b + c + bc \right)\]
\[ = a + b + c + bc + ab + ac + abc\]
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\]
\[ = a + b + ab + c + \left( a + b + ab \right) c\]
\[ = a + b + c + ab + ac + bc + abc\]
\[\text{ Therefore },\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in N\]
Thus, * is associative on N.
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