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Question
Determine which of the following binary operation is associative and which is commutative : * on N defined by a * b = 1 for all a, b ∈ N ?
Solution
Commutativity :
\[\text{Let a}, b \in N . \text{Then}, \]
\[a * b = 1 \]
\[b * a = 1\]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity :
\[\text{ Let } a, b, c \in N . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( 1 \right)\]
\[ = 1\]
\[\left( a * b \right) * c = \left( 1 \right) * c\]
\[ = 1\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in N\]
Thus, * is associative on N .
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