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 R defined by ab = a + b – 7 for all a,b∈Q.
Answer» Given that * is a binary operation on R defined by ab = a + b – 7 for all a,b∈R. 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 – 7 ⇒ ba = b + a – 7 = a + b – 7 ⇒ ba = ab ∴ Commutative property holds for given binary operation ‘’ on ‘R’. We know that associative property is (pq)r = p(qr)* Let’s check the associativity of given binary operation: ⇒ (ab)c = (a + b – 7)c ⇒ (ab)c = a + b – 7 + c – 7 ⇒ (ab)c = a + b + c – 14 ...... (1) ⇒ a(bc) = a(b + c – 7) ⇒ a(bc) = a + b + c – 7 – 7 ⇒ a(bc) = a + b + c – 14 ...... (2) *From (1) and (2) we can clearly say that associativity holds for the binary operation ‘’ on ‘R’.**

Check the commutativity and associativity of each of the following binary operations: ‘’ on R defined by ab = a + b – 7 for all a,b∈Q.

Answer»

Given that * is a binary operation on R defined by a*b = a + b – 7 for all a,b∈R.

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 – 7

⇒ b*a = b + a – 7 = a + b – 7

⇒ b*a = a*b

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

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 – 7)*c

⇒ (a*b)*c = a + b – 7 + c – 7

⇒ (a*b)*c = a + b + c – 14 ...... (1)

⇒ a*(b*c) = a*(b + c – 7)

⇒ a*(b*c) = a + b + c – 7 – 7

⇒ a*(b*c) = a + b + c – 14 ...... (2)

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