Question
Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all $\text{a, b}\in\text{N}.$
Check the commutativity and associativity of '*' on N.
Check the commutativity and associativity of '*' on N.
a * b = 1.c.m. a, b
= 1.c.m. b, a
= b * a
Therefore,
$\text{a}\ ^*\ \text{b}=\text{b}\ ^*\ \text{a}\ \forall\ \text{a, b}\in\text{N}$
Thus, * is Commutative on N.
Associativity: Let $\text{a, b, c}\in\text{N}$
a * b * c = a * 1.c.m. b, c
= 1.c.m. a, b, c
a * b * c = 1.c.m. a, b * c
= 1.c.m. a, b, c
Therefore,
$\text{a}\ ^*\ \text{b}\ ^*\ \text{c}=\text{a}\ ^*\ \text{b}\ ^*\ \text{c}\ \forall\ \text{a, b, c}\in\text{N}$
Thus, * is associative on N.
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