Check the commutativity and associativity of each of the following bin

1.	Check the commutativity and associativity of each of the following binary operations: ‘’ on Z defined by ab = a + b – ab for all a,b∈Z
Answer» Given that * is a binary operation on Z defined by ab = a + b – ab for all a,b∈Z. We know that commutative property is pq = qp, where is a binary operation.** Let’s check the commutativity of given binary operation: ⇒ ab = a + b – ab ⇒ ba = b + a – ba = a + b – ab ⇒ ba = ab ∴ Commutative property holds for given binary operation ‘’ on ‘Z’. We know that associative property is (pq)r = p(qr)* Let’s check the associativity of given binary operation: ⇒ (ab)c = (a + b – ab)c ⇒ (ab)c = a + b – ab + c – ((a + b – ab)×c) ⇒ (ab)c = a + b + c – ab – ac – bc + abc ...... (1) ⇒ a(bc) = a(b + c – bc) ⇒ a(bc) = a + b + c – bc – (a×(b + c – bc)) ⇒ a(bc) = a + b + c – ab – ac – bc + abc ...... (2) *From (1) and (2) we can clearly say that associativity hold for the binary operation ‘’ on ‘Z’.**

Check the commutativity and associativity of each of the following binary operations: ‘’ on Z defined by ab = a + b – ab for all a,b∈Z

Answer»

Given that * is a binary operation on Z defined by a*b = a + b – ab for all a,b∈Z.

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 + b – ab

⇒ b*a = b + a – ba = a + b – ab

⇒ b*a = a*b

∴ Commutative property holds 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 + b – ab)*c

⇒ (a*b)*c = a + b – ab + c – ((a + b – ab)×c)

⇒ (a*b)*c = a + b + c – ab – ac – bc + abc ...... (1)

⇒ a*(b*c) = a*(b + c – bc)

⇒ a*(b*c) = a + b + c – bc – (a×(b + c – bc))

⇒ a*(b*c) = a + b + c – ab – ac – bc + abc ...... (2)

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