Let
*
be the binary operation on N
given 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 *?
Let * be the binary operation on N given 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 *?
The binary
operation *
on N is defined as a *
b = L.C.M. of a and b.
(i) 5 *
7 = L.C.M. of 5 and 7 = 35
20 *
16 = L.C.M of 20 and 16 = 80
(ii) It is
known that:
L.C.M of a and b = L.C.M of b and a &mnForE;
a, b ∈ N.
∴a *
b = b *
a
Thus, the operation *
is commutative.
(iii) For
a, b, c ∈ N,
we have:
(a *
b) *
c = (L.C.M of a and b) *
c = LCM of a, b, and c
a *
(b *
c) = a *
(LCM of b and c) = L.C.M of a, b, and c
∴(a *
b) *
c = a *
(b *
c)
Thus, the operation *
is associative.
(iv) It is
known that:
L.C.M. of a and 1 = a = L.C.M. 1 and a &mnForE;
a ∈ N
⇒ a *
1 = a = 1 *
a &mnForE; a ∈
N
Thus, 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.
Here, e
= 1
This means that:
L.C.M of a and b = 1 = L.C.M of b and a
This case is possible only when a and b are equal to 1.
Thus, 1 is the only invertible element of N with respect to
the operation *.
The binary operation * on N is defined as a * b = L.C.M. of a and b.
(i) 5 * 7 = L.C.M. of 5 and 7 = 35
20 * 16 = L.C.M of 20 and 16 = 80
(ii) It is known that:
L.C.M of a and b = L.C.M of b and a &mnForE; a, b ∈ N.
∴a * b = b * a
Thus, the operation * is commutative.
(iii) For a, b, c ∈ N, we have:
(a * b) * c = (L.C.M of a and b) * c = LCM of a, b, and c
a * (b * c) = a * (LCM of b and c) = L.C.M of a, b, and c
∴(a * b) * c = a * (b * c)
Thus, the operation * is associative.
(iv) It is known that:
L.C.M. of a and 1 = a = L.C.M. 1 and a &mnForE; a ∈ N
⇒ a * 1 = a = 1 * a &mnForE; a ∈ N
Thus, 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.
Here, e = 1
This means that:
L.C.M of a and b = 1 = L.C.M of b and a
This case is possible only when a and b are equal to 1.
Thus, 1 is the only invertible element of N with respect to the operation *.
