Let * be the binary operation on N defined by a * b = H.C.F. of a and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N?

Solution:

The commutative property deals with the arithmetic operations of addition and multiplication. It means that changing the order or position of numbers while adding or multiplying them does not change the end result.

The binary operation * on N is defined by a * b

= HCF of a and b.

⇒ a * b = b * a

Operation * is commutative.

For all a, b, c ∈ N,

(a * b) * c = (HCF of a and b) * c

= HCF of a, b, c

a * (b * c) = a * (HCF. of b and c)

= HCF of a, b, c

Therefore,

(a * b) * c = a * (b * c)

Operation * is associative.

e ∈ N will be the identity for the operation* if a * e = a = e * a for all a ∈ N.

But this relation is not true for any a ∈ N.

Operation * does not have any identity in N

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.4 Question 8

Let * be the binary operation on N defined by a * b = H.C.F. of a and b. Is * commutative? Is * associative? Does there exist an identity for this binary operation on N?

Summary:

For the binary operation on N defined by a * b = H.C.F. of a and b is commutative and associative. Operation * does not have any identity in N