# Can a DFA have an empty string as input?

Given string w such that every odd position in w is a 1

My solution is

The book's solution is

from my understanding $\epsilon$ should not be an accepted state, however in the book's solution it is. How is it handling this case?

In this case, the argument is correct. Think of the problem statement as an implication. If $i$ is an odd position, then $S_i$ is 1.