Intersection of two incomplete DFAs
To answer the question, there are two different manners to solve this.


By adding a dead state whenever you have an incomplete automata. By doing so, you will no longer have to worry about what to do if one automata has a transition from a specific state but not the other.
You can still do the intersection of the two without adding dead states.




In this example, if you are focusing on the state 1 and 3 with the symbol 'b' :
- From A1, you have no transition : δ(1,b) = ∅
- From A2, you have a transition : δ(3,b) = 4

Since one has a transition, but not the other, then the resulting automata will have the transition : δ(13,b) = ∅. Therefore, it is indeed acceptable to put ∅.