# Problems with simulating NFA with DFA

I have this NFA:

I want to simulate it with a DFA, thus I have the states {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}. Usually I look at every state and find out in which state I could end up when selecting a letter. However here the problem starts at {1}, where the dfa wouldn't have any connection for letter a and thus wouldn't be a proper DFA. How can I resolve this ?

You missed state $$\emptyset$$. The transition from state $$1$$ when character $$a$$ is read goes to $$\emptyset$$.
Since all transition from $$\emptyset$$ also go to $$\emptyset$$, you can also use the standard convention of omitting $$\emptyset$$ (and the transition for character $$a$$ from state $$1$$).
If a DFA has some missing transitions, it can always be completed by first adding a non-final state $$q$$, and then adding all missing transitions to $$q$$.