Q. 105.0( 1 Vote )

# On Z, the set of

Given that * is a binary operation on Z defined by a*b = a + 3b – 4 for all a,bZ.

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 = a + 3b – 4

b*a = b + 3a – 4

b*a ≠ a*b

The commutative property doesn’t hold for given binary operation * on Z.

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

Let’s check the associativity of given binary operation:

(a*b)*c = (a + 3b – 4)*c

(a*b)*c = a + 3b – 4 + 3c – 4

(a*b)*c = a + 3b + 3c – 8 ...... (1)

a*(b*c) = a*(b + 3c – 4)

a*(b*c) = a + (3×(b + 3c – 4)) – 4

a*(b*c) = a + 3b + 9c – 12 – 4

a*(b*c) = a + 3b + 9c – 16 ...... (2)

From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘Z’.

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