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Question
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.
Solution
Commutativity:
\[\text{Let }a, b \in Q - \left\{ - 1 \right\} . \text{Then}, \]
\[a * b = a + b + ab\]
\[ = b + a + ba\]
\[ = b * a\]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in Q - \left\{ - 1 \right\}\]
Thus, * is commutative on Q - {-1}
Associativity:
\[\text{Let }a, b, c \in Q - \left\{ - 1 \right\} . \text{ Then }, \]
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\]
\[ = a + b + c + bc + a \left( b + c + bc \right)\]
\[ = a + b + c + bc + ab + ac + abc\]
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\]
\[ = a + b + ab + c + \left( a + b + ab \right)c\]
\[ = a + b + c + ab + ac + bc + abc\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q - \left\{ - 1 \right\} . \]
Thus, * is associative on Q -{-1]
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