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