Question

The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ________________ .

विकल्प

• commutative only

• associative only

• commutative and associative both

• none of these

Answer 1

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.