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