Q. 4 O5.0( 1 Vote )

Check the commuta

Answer :

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

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’.

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