# DFA that recognizes 0 in every odd position

I had a question that asked for the DFA that accepts the following language:

{w | every odd position of w is a 0 }


Should this DFA accept the empty string, since it doesn't have an odd position?

• "Should this DFA accept the empty string, since it doesn't have an odd position?" - Yes. The condition vacuously holds. – Guildenstern Mar 18 '15 at 13:46
• $\forall x \in \emptyset : P(x)$ holds for all predicates $P$. – Raphael Mar 18 '15 at 14:19

Yes, since $|\epsilon| = 0$, there are no odd positions; therefore, that DFA would accept $\epsilon$.