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Check the Commutativity and Associativity of the Following Binary Operations '*'. On N Defined By A * B = 2ab For All A, B ∈ N ? - Mathematics

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प्रश्न

Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?

योग
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उत्तर

 Commutativity:

\[\text{Let } a, b \in N . \text{Then}, \]

\[a * b = 2^{ab} \]

\[ = 2^{ba} \]

\[ = b * a\]

\[\text{Therefore},\]

\[a * b = b * a, \forall a, b \in N\]

Thus, * is commutative on N.

Associativity:

\[\text{Let a}, b, c \in N . \text{Then}, \]

\[a * \left( b * c \right) = a * \left( 2^{bc} \right)\]

\[ = 2^{a * 2^{bc}} \]

\[\left( a * b \right) * c = \left( 2^{ab} \right) * c\]

\[ = 2^{ab * 2^c} \]

\[\text{Therefore},\]

\[a * \left( b * c \right) \neq \left( a * b \right) * c\]


Thus, * is not associative on N.

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अध्याय 3: Binary Operations - Exercise 3.2 [पृष्ठ १२]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 3 Binary Operations
Exercise 3.2 | Q 4.02 | पृष्ठ १२

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