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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.
Solution
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.
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