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.

Answer 1

*	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

$$\begin{gathered} LetS=\left\{1,2,3,4,5\right\} \\ Sincea*balwayslieinS,therefore \\ *isamapfromS\times StoS \\ Henceitisabinaryoperation \\ Foridentityelementetoexist \\ a*e=e*a=a,foralla\in S \\ Fromcompositiontablewecanseesuchanelementdoesnotexist. \\ Identityelementdoesnotexist \\ a*b=H.C.F\left(a,b\right)=H.C.F\left(b,a\right)=b*a \\ Hence*iscommutative \\ \left(a*b\right)*c=\left(H.C.F.\left(a,b\right)\right)*c=H.C.F.\left(\left(H.C.F.\left(a,b\right)\right),c\right)=H.C.F.\left(a,b,c\right)=H.C.F.\left(c,a,b\right)=H.C.F.\left(b,\left(H.C.F.\left(a,c\right)\right)\right)=b*\left(a*c\right) \\ Hence*isassociative \end{gathered}$$