Question

Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N

Check the commutativity and associativity of '*' on N.

Answer 1

Commutativity :

\[\text{Let a}, b \in N\] 
\[a * b = 1.c.m.\left( \text{a, b} \right)\] 
\[ =1.c.m].\left( \text{b, a} \right)\] 
\[ =\text{b * a}\] 
\[\text{Therefore},\] 
\[a * b = b * a, \forall a, b \in N\]

Thus, * is commutative on N.

Associativity:

\[\text{Let a}, \text{b, c }\in N\] 
\[a * \left( b * c \right) = a * 1.c.m.\left( \text{b, c} \right)\] 
\[ = 1.c.m.\left( a, \left( \text{b, c} \right) \right)\] 
\[ = 1.c.m.\left( \text{ a, b, c} \right)\] 
\[\left( a * b \right) * c = 1.c.m.\left( a, b \right)* c\] 
\[ = 1.c.m.\left( \left( \text{a, b} \right), c \right)\] 
\[ = 1.c.m.\left( a, b, c \right)\] 
\[\text{Therefore},\] 
\[a * \left( \text{b * c} \right) = \left( \text{a * b}  \right) * c, \forall \text{a, b, c} \in N\]

Thus, * is associative on N.