Check the commutativity and associativity of each of the following binary operations:
‘*’ on Q defined by a*b = gcd(a,b) for all a,b∈Q

Question

Check the commutativity and associativity of each of the following binary operations: ‘*’ on Q defined by a*b = gcd(a,b) for all a,b ∈ Q

Philoid · Accepted Answer

Given that * is a binary operation on Q defined by a*b = g.c.d(a,b) for all a,b∈Q.

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 = g.c.d(a,b)

⇒ b*a = g.c.d(b,a) = g.c.d(a,b)

⇒ b*a = a*b

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

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

Let’s check the associativity of given binary operation:

⇒ (a*b)*c = (g.c.d(a,b))*c

⇒ (a*b)*c = g.c.d(g.c.d(a,b),c)

⇒ (a*b)*c = g.c.d(a,b,c) ...... (1)

⇒ a*(b*c) = a*(g.c.d(b,c))

⇒ a*(b*c) = g.c.d(a,g.c.d(b,c))

⇒ a*(b*c) = g.c.d(a,b,c) ...... (2)

From (1) and (2) we can clearly say that associativity hold for the binary operation ‘*’ on ‘Q’.

Check the commutativity and associativity of each of the following binary operations:‘*’ on Q defined by a*b = gcd(a,b) for all a,b∈Q