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Question
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].
Show that 'o' is both commutative and associate ?
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Solution
Commutativity :
\[\text{ Let }a, b \in Q_0 . \text{Then}, \]
\[a o b = \frac{ab}{2}\]
\[ = \frac{ba}{2}\]
\[ = b o a\]
\[\text{Therefore},\]
\[a o b = b o a, \forall a, b \in Q_0\]
Thus, o is commutative on Qo.
Associativity:
\[\text{ Let }a, b, c \in Q_0 . \text{ Then }, \]
\[a o \left( b o c \right) = a o \left( \frac{bc}{2} \right)\]
\[ = \frac{a\left( \frac{bc}{2} \right)}{2}\]
\[ = \frac{abc}{4}\]
\[\left( a o b \right) o c = \left( \frac{ab}{2} \right) o c\]
\[ = \frac{\left( \frac{ab}{2} \right)c}{2}\]
\[ = \frac{abc}{4}\]
\[\text{Therefore},\]
\[a o \left( b o c \right) = \left( a o b \right) o c, \forall a, b, c \in Q_0 \]
Thus, o is associative on Qo.
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