Question

 Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?

Answer 1

Commutativity :

\[\text{Let a}, b \in Q . \text{Then}, \]

\[a * b = ab + 1\]

      \[ = ba + 1\]

       \[ = b * a \]

\[\text{Therefore},\]

\[a * b = b * a, \forall a, b \in Q\]

Thus, * is commutative on Q.

Associativity:

\[\text{Let } a, b, c \in Q . \text{Then}, \]

\[a * \left( b * c \right) = a * \left( bc + 1 \right)\]

\[ = a\left( bc + 1 \right) + 1\]

\[ = abc + a + 1\]

\[\left( a * b \right) * c = \left( ab + 1 \right) * c\]

\[ = \left( ab + 1 \right)c + 1\]

\[ = abc + c + 1\]

\[\text{Therefore},\]

\[a * \left( b * c \right) \neq \left( a * b \right) * c\]

Thus, * is not associative on Q.