Let * be the operation on set {1,2,3,4,5} defined by a*b = H C F ( a - Maths - Relations and Functions

Question

Let * be the operation on set {1,2,3,4,5} defined by a*b = H C F (
a & b ) Form composition table and state whether the operation
is binary or not Also, check for commutative & assosciative
properties Does identity element exist ? Justify

Aarushi Mishra · Accepted Answer

*
1
2
3
4
5

1
1
1
1
1
1

2
1
2
1
2
1

3
1
1
3
1
1

4
1
2
1
4
1

5
1
1
1
1
5

Let S=1, 2, 3, 4, 5Since a*b always lie in S, therefore * is a map from S×S to SHence it is a binary operationFor identity element e to exista*e=e*a=a, for all a∈SFrom composition table we can see such an element doesnot exist
Identity element doesnot exista*b=H C F a,b=H C
F b,a=b*aHence *is commutativea*b*c=H C F
 a,b*c=H C F  H C F  a,b, c=H C F
 a, b, c=H C F  c, a, b=H C F
b, H C F  a,c=b*a*cHence * is associative

Let * be the operation on set {1,2,3,4,5} defined by a*b = H.C.F. ( a & b ). Form composition table and state whether the operation is binary or not. Also, check for commutative & assosciative properties. Does identity element exist ? Justify.