Let ‘*’ be a binary operation on N defined by a*b = L.C.M(a,b) for

1.	Let ‘’ be a binary operation on N defined by ab = L.C.M(a,b) for all a,b∈N. Check the commutativity and associativity of ‘*’ on N.
Answer» *We know that commutative property is pq = qp, where is a binary operation.** Let’s check the commutativity of given binary operation: ⇒ ab = L.C.M(a,b) ⇒ ba = L.C.M(b,a) = L.C.M(a,b) ⇒ ba = ab ∴ Commutative property holds for given binary operation ‘’ on ‘N’. We know that associative property is (pq)r = p(qr) Let’s check the associativity of given binary operation: ⇒ (ab)c = (L.C.M(a,b))c ⇒ (ab)c = L.C.M(a,b)c ⇒ (ab)c = L.C.M(L.C.M(a,b),c) ⇒ (ab)c = L.C.M(a,b,c) ...... (1) ⇒ a(bc) = a(L.C.M(b,c)) ⇒ a(bc) = aL.C.M(b,c) ⇒ a(bc) = L.C.M(a,L.C.M(b,c)) ⇒ a(bc) = L.C.M(a,b,c) ...... (2) From(1) and (2) we can say that associative property holds for binary function ‘’ on ‘N’**

Let ‘’ be a binary operation on N defined by ab = L.C.M(a,b) for all a,b∈N. Check the commutativity and associativity of ‘*’ on N.

Answer»

We know that commutative property is p*q = q*p, where * is a binary operation.

Let’s check the commutativity of given binary operation:

⇒ a*b = L.C.M(a,b)

⇒ b*a = L.C.M(b,a) = L.C.M(a,b)

⇒ b*a = a*b

∴ Commutative property holds for given binary operation ‘*’ on ‘N’.

We know that associative property is (p*q)*r = p*(q*r)

Let’s check the associativity of given binary operation:

⇒ (a*b)*c = (L.C.M(a,b))*c

⇒ (a*b)*c = L.C.M(a,b)*c

⇒ (a*b)*c = L.C.M(L.C.M(a,b),c)

⇒ (a*b)*c = L.C.M(a,b,c) ...... (1)

⇒ a*(b*c) = a*(L.C.M(b,c))

⇒ a*(b*c) = a*L.C.M(b,c)

⇒ a*(b*c) = L.C.M(a,L.C.M(b,c))

⇒ a*(b*c) = L.C.M(a,b,c) ...... (2)

From(1) and (2) we can say that associative property holds for binary function ‘*’ on ‘N’

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