Let * be the binary operation on N defined by a * b = L.C.M. of a and b. Find
i. 5 * 7, 20 * 16
ii. Is * commutative?
iii. Is * associative?
iv. Find the identity of * in N
v. Which elements of N are invertible for the operation *?

Solution:

The binary operation on N is defined by a * b = L.C.M. of a and b.

(i). 5 * 7 = L.C.M of 5 and 7 = 35

20 * 16 = LCM of 20 and 16 = 80

(ii). L.C.M. of a and b

= LCM of b and a for all a, b ∈ N

⇒ a * b = b * a

Operation * is commutative.

(iii). For a, b, c ∈ N

(a * b) * c = (L.C.M. of a and b) * c

= L.C.M. of a, b, c

a * (b * c) = a * (L.C.M. of b and c)

= L.C.M. of a, b, c

⇒ (a * b) * c = a * (b * c)

Operation * is associative.

(iv). L.C.M. of a and 1= a = L.C.M. of 1 and a for all a ∈ N

a * 1 = a = 1 * a for all a Î N

Therefore, 1 is the identity of * in N.

(v). An element, a in N is invertible with respect to the operation * if there exists an element b in N,

such that a * b = e = b * a

e = 1

L.C.M. of a and b = 1

= LCM of b and a possible only when a and b are equal to 1.

1 is the only invertible element of N with respect to the operation *

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.4 Question 6

Let * be the binary operation on N defined by a * b = L.C.M. of a and b. Find (i). 5 * 7, 20 * 16 (ii). Is * commutative? (iii). Is * associative? (iv). Find the identity of * in N (v). Which elements of N are invertible for the operation *?

Summary:

For the given binary operation on N defined by a * b = L.C.M. of a and b.  Operation * is commutative and associative. Therefore, 1 is the identity of * in N. 1 is the only invertible element of N with respect to the operation *