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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 *
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 *
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