Question

Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?

Answer 1

 Commutativity :

\[\text{ Let } a, b \in N . \text{Then}, \] 
\[a * b = \frac{a + b}{2}\] 
\[ = \frac{b + a}{2}\] 
\[ = b * a\] 
\[\text{Therefore},\] 
\[a * b = b * a, \forall a, b \in N\]

Thus, * is commutative on N.

Associativity:

\[\text{Let }a, b, c \in N . \text{Then}, \] 
\[a * \left( b * c \right) = a * \left( \frac{b + c}{2} \right)\] 
\[ = \frac{a + \left( \frac{b + c}{2} \right)}{2}\] 
\[ = \frac{2a + b + c}{4}\]
\[\left( a * b \right) * c = \left( \frac{a + b}{2} \right) * c\] 
\[ = \frac{\left( \frac{a + b}{2} \right) + c}{2}\] 
\[ = \frac{a + b + 2c}{4}\] 
\[\text{Thus},a * \left( b * c \right) \neq \left( a * b \right) * c\] 
\[\text{ If a} = 1, b = 2, c = 3\] 
\[1 * \left( 2 * 3 \right) = 1 * \left( \frac{2 + 3}{2} \right)\] 
\[ = 1 * \frac{5}{2}\] 
\[ = \frac{1 + \frac{5}{2}}{2}\] 
\[ = \frac{7}{4}\] 
\[\left( 1 * 2 \right) * 3 = \left( \frac{1 + 2}{2} \right) * 3\] 
\[ = \frac{3}{2} * 3\] 
\[ = \frac{\frac{3}{2} + 3}{2}\] 
\[ = \frac{9}{4}\] 
\[\text { Therefore, }\exists  \text{ a} = 1, b = 2, c = 3 \in \text{ N such that a}  * \left( b * c \right) \neq \left( a * b \right) * c\]

Thus, * is not associative on N.