Determine which of the following binary operations are associative and

1.	Determine which of the following binary operations are associative and which are commutative: * on N defined by a*b = 1 for all a,b N
Answer» Given that * is a binary operation on N defined by ab = 1 for all a,b∈N. We know that commutative property is pq = qp, where is a binary operation.** Let’s check the commutativity of given binary operation: ⇒ ab = 1 ⇒ ba = 1 ⇒ ba = ab ∴ The commutative property holds for given binary operation ‘’ on ‘N’. We know that associative property is (pq)r = p(qr)* Let’s check the associativity of given binary operation: ⇒ (ab)c = (1)c ⇒ (ab)c = 1c ⇒ (ab)c = 1 ...... (1) ⇒ a(bc) = a(1) ⇒ a(bc) = a1 ⇒ a(bc) = 1 ...... (2) From (1) and (2) we can clearly say that, *Associative property holds for given binary operation ‘’ on ‘N’**

Determine which of the following binary operations are associative and which are commutative: * on N defined by a*b = 1 for all a,b N

Answer»

Given that * is a binary operation on N defined by a*b = 1 for all a,b∈N.

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 = 1

⇒ b*a = 1

⇒ b*a = a*b

∴ The commutative property holds for given binary operation ‘*’ on ‘N’.

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

Let’s check the associativity of given binary operation:

⇒ (a*b)*c = (1)*c

⇒ (a*b)*c = 1*c

⇒ (a*b)*c = 1 ...... (1)

⇒ a*(b*c) = a*(1)

⇒ a*(b*c) = a*1

⇒ a*(b*c) = 1 ...... (2)

From (1) and (2) we can clearly say that,

Associative property holds for given binary operation ‘*’ on ‘N’

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