Q. Let R be a binary relation on $A=\{a,b,c,d,e,f,g,h\}$ represented by the following two component digraph.
Component 1 - cycle of length 3.
Component 2 - cycle of length 5.
Find the smallest integers $m \ and \ n$ such that $ m<n \ and \ R^m=R^n.$
My approach- If a relation has a cycle of length n then its $n^{th}$ power is reflexive. So for the $1^{st}$ component $R^3$ is reflexive hence $R^1=R^4$. Similarly for the $2^{nd}$ component $R^1=R^6$. LCM of 4 and 6 is 12.. So m=1 and n=12.
But somewhere I found the answer is m= 0 and n=15
I don't know where I am wrong. Please help me.
By the same thought, restricted to the first component, we have $R^3=id$, and $R^5=id$ when restricted to the other one. So $R^{15}=id=R^0$.