Let ‘*’ be a binary operation on N defined by a * b = l.c.m (

1.	Let ‘’ be a binary operation on N defined by a b = l.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» (i) Given as a * b = 1.c.m. (a, b) 2 * 4 = l.c.m. (2, 4) = 4 3 * 5 = l.c.m. (3, 5) = 15 1 * 6 = l.c.m. (1, 6) = 6 (ii) Let us prove commutativity of * Let a, b ∈ N a * b = l.c.m (a, b) = l.c.m (b, a) = b * a So a * b = b * a ∀ a, b ∈ N Thus * is commutative on N. Now let us prove associativity of * Let a, b, c ∈ N a * (b * c ) = a * l.c.m. (b, c) = l.c.m. (a, (b, c)) = l.c.m (a, b, c) (a * b) * c = l.c.m. (a, b) * c = l.c.m. ((a, b), c) = l.c.m. (a, b, c) So (a * (b * c) = (a * b) * c, ∀ a, b , c ∈ N Hence, * is associative on N.

Let ‘’ be a binary operation on N defined by a b = l.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»

(i) Given as a * b = 1.c.m. (a, b)

2 * 4 = l.c.m. (2, 4)

= 4

3 * 5 = l.c.m. (3, 5)

= 15

1 * 6 = l.c.m. (1, 6)

= 6

(ii) Let us prove commutativity of *

Let a, b ∈ N

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

= l.c.m (b, a)

= b * a

a * b = b * a ∀ a, b ∈ N

Thus * is commutative on N.

Now let us prove associativity of *

Let a, b, c ∈ N

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

= l.c.m. (a, (b, c))

= l.c.m (a, b, c)

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

= l.c.m. ((a, b), c)

= l.c.m. (a, b, c)

(a * (b * c) = (a * b) * c, ∀ a, b , c ∈ N

Hence, * is associative on N.