Consider a set a = {1,2,3,6} and let * be binary operation on a defined by a*b=1cm of a and b. check whether * is commutative and associative. find the identity element if exists. write another commutative binary operation on a with 2 as the identity
Answers
Binary operation * is commutative and associative and its identity element exists
The identity element is 1
Step-by-step explanation:
Given
A set A = {1,2,3,6}
A binary operation on A is defined as
a*b = LCM of a and b
(1) Commutativity Check
* is commutative if
a*b = b*a
a*b = LCM of a and b
b*a = LCM of b and a = LCM of a and b
Thus
a*b = b*a
Therefore, the binary operation * is commutative
(2) Associativity Check
* is associative if
(a*b)*c = a*(b*c)
(a*b)*c)
= (LCM of a and b)*c
= LCM of a and b and LCM of c
= LCM of a and b and c
a*(b*c)
= LCM of a and b*c
= LCM of a and (LCM of b and c)
= LCM of a and b and c
Thus
(a*b)*c = a*(b*c)
Therefore, the binary operation * is associative
If e is the identity element then
a*e = e*a = a
If e = 1
Then a*e = LCM of a and 1 = a
And e*a = LCM of 1 and a = a
Therefore, the identity element is 1
If we define a binary operation such that
a*b = (a×b)/2
Then 2 will be the identity element of this binary operation
Hope this answer is helpful.
Know More:
Q: 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 *?
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