Question

Let '*' be a binary operation on N defined by
a * b = 1.c.m. (a, b) for all a, b ∈ N
(i) Find 2 * 4, 3 * 5, 1 * 6.
(ii) Check the commutativity and associativity of '*' on N.

Answer 1

a * b = 1.c.m. (a, b)

(i) 2 * 4 = 1.c.m. (2, 4)
= 4
3 * 5 = 1.c.m. (3, 5)
= 15
1 * 6 = 1.c.m. (1, 6)
= 6

(ii) Commutativity:
$$\begin{gathered} \text{Let}a,b\in N \\ a*b=1\text{.c.m.}\left(a,b\right) \\ \text{=1.c.m.}\left(b,a\right) \\ \text{=}b*a \\ \text{Therefore,} \\ a*b=b*a,\forall a,b\in N \end{gathered}$$

Thus, * is commutative on N.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in N \\ a*\left(b*c\right)=a*1\text{.c.m.}\left(b,c\right) \\ \text{= 1.c.m.}\left(a,\left(b,c\right)\right) \\ \text{= 1.c.m.}\left(a,b,c\right) \\ \left(a*b\right)*c=1\text{.c.m.}\left(a,b\right)\text{* c} \\ \text{= 1.c.m.}\left(\left(a,b\right),c\right) \\ \text{= 1.c.m.}\left(a,b,c\right) \\ \text{Therefore,} \\ a*\left(b*c\right)=\left(a*b\right)*c,\forall a,b,c\in N \end{gathered}$$

Thus, * is associative on N.