Let A be any set containing more than one element. Let ‘*’ be a bi

1.	Let A be any set containing more than one element. Let ‘’ be a binary operation on A defined by ab = b for all a,b∈A. Is ‘*’ commutative or associative on A?
Answer» Given that * is a binary operation on set A defined by ab = b for all a,b∈A. We know that commutative property is pq = qp, where is a binary operation.** Let’s check the commutativity of given binary operation: ⇒ ab = b ⇒ ba = a ⇒ ba≠ab ∴ The commutative property does not hold for given binary operation ‘’ on ‘A’. We know that associative property is (pq)r = p(qr)* Let’s check the associativity of given binary operation: ⇒ (ab)c = (b)c ⇒ (ab)c = bc ⇒ (ab)c = c ...... (1) ⇒ a(bc) = a(c) ⇒ a(bc) = ac ⇒ a(bc) = c ...... (2) *From (1) and (2) we can clearly say that associativity holds for the binary operation ‘’ on ‘A’.**

Let A be any set containing more than one element. Let ‘’ be a binary operation on A defined by ab = b for all a,b∈A. Is ‘*’ commutative or associative on A?

Answer»

Given that * is a binary operation on set A defined by a*b = b for all a,b∈A.

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 = b

⇒ b*a = a

⇒ b*a≠a*b

∴ The commutative property does not hold for given binary operation ‘*’ on ‘A’.

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

Let’s check the associativity of given binary operation:

⇒ (a*b)*c = (b)*c

⇒ (a*b)*c = b*c

⇒ (a*b)*c = c ...... (1)

⇒ a*(b*c) = a*(c)

⇒ a*(b*c) = a*c

⇒ a*(b*c) = c ...... (2)

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

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