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Question
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
Solution
\[\text{Let }a, b, c \in Q . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( \frac{bc}{7} \right)\]
\[ = \frac{a\left( \frac{bc}{7} \right)}{7}\]
\[ = \frac{abc}{49}\]
\[\left( a * b \right) * c = \left( \frac{ab}{7} \right) * c\]
\[ = \frac{\left( \frac{ab}{7} \right)c}{7}\]
\[ = \frac{abc}{49}\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q\]
Thus, * is associative on Q.
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