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 Q defined by ab = a + ab for all a,b∈Q
Answer» Given that * is a binary operation on Q defined by ab = a + ab for all a,b∈Q. We know that commutative property is pq = qp, where is a binary operation.** Let’s check the commutativity of given binary operation: ⇒ ab = a + ab ⇒ ba = b + ba = b + ab ⇒ ba≠ab ∴ Commutative property doesn’t holds for given binary operation ‘’ on ‘Q’. We know that associative property is (pq)r = p(qr)* Let’s check the associativity of given binary operation: ⇒ (ab)c = (a + ab)c ⇒ (ab)c = a + ab + ((a + ab)×c) ⇒ (ab)c = a + ab + ac + abc ...... (1) ⇒ a(bc) = a(b + bc) ⇒ a(bc) = a + (a×(b + bc)) ⇒ a(bc) = a + ab + abc ...... (2) *From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘’ on ‘Q’.**

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

Answer»

Given that * is a binary operation on Q defined by a*b = a + ab 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 = a + ab

⇒ b*a = b + ba = b + ab

⇒ b*a≠a*b

∴ Commutative property doesn’t 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 = (a + ab)*c

⇒ (a*b)*c = a + ab + ((a + ab)×c)

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

⇒ a*(b*c) = a*(b + bc)

⇒ a*(b*c) = a + (a×(b + bc))

⇒ a*(b*c) = a + ab + abc ...... (2)

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

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