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 N defined by ab = 2ab for all a,b∈N.
Answer» Given that * is a binary operation on N defined by ab = 2^ab 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 = 2^ab ⇒ ba = 2^ba = 2^ab ⇒ 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 = (2^ab)c ⇒ \((a b ) * c = 2^{ab}.c\) ..........(1) ⇒ a(bc) = a(2^bc) ⇒ \((a b ) * c = 2^{a.2^{bc}}\) ...........(2) *From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘’ on ‘N’.**

Check the commutativity and associativity of each of the following binary operations: ‘’ on N defined by ab = 2ab for all a,b∈N.

Answer»

Given that * is a binary operation on N defined by a*b = 2^ab 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 = 2^ab

⇒ b*a = 2^ba = 2^ab

⇒ 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 = (2^ab)*c

⇒ \((a * b ) * c = 2^{ab}.c\) ..........(1)

⇒ a*(b*c) = a*(2^bc)

⇒ \((a * b ) * c = 2^{a.2^{bc}}\) ...........(2)

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

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