Search on Conformant Problem: solution for subset of a belief state
'different path from the start state': The sensorless vacuum world is by definition a 'single-state problem', that is, all variables in the world are incorporated into a single state variable. In that case, history doesn't matter. If we are in state 5 by one path, the world is exactly as it would be if we had got there by another path.

For an example of something else represented as a single-state problem, consider chess: you might say history matters because whether you can castle, depends on whether the king has moved, not just the current board position. Okay so if we represent chess as a single-state problem, then the state would encode not only the current board position but also other relevant variables like whether the king has moved.

Subsets: if we know how to get to {5,7}, we don't need to be interested in {1,3,5,7}. Getting to {5,7} gives us all the same information plus more besides.