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 ?

Answer 1

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 .