# Is it possible to OR with an empty set?

I am working with converting an NFA to a DFA and came across an odd set notation issue that I don't know how to answer.

Say I have the following NFA and assume the starting state to be zero:

So if I let q0 = A, q1 = B, q2 = C then B is my only accepted state.

Looking at all 3 states with transitions 0 and 1 I find transition 0 to be the following

A -> {A,B}

B -> {C}

C -> {emptyset}

With Transition 1 I get the following:

A -> {B}

B -> {C}

C -> {C}

so in my DFA A takes me to state {AB} an accepting state with 1 or {B} also an accepting state with 0 and I need to carry on to minimize the states.

Eventually I end up with the following with transition 0:

{ABC} -> {A,B,C,NULL}.

Can I translate this into $$A$$ or $$B$$ or $$C$$ or $$\emptyset$$?

• Some additional context would be helpful. Feb 16, 2017 at 18:01
• I added the actual problem in addition to the support provided by John's edits. @MichaelBurr I can attach my steps and a picture of where I am currently stuck as well if that helps. Feb 16, 2017 at 19:13

I believe you must have encountered a state which has no transition specified for a particular input symbol. As for your question yes you can take the union(OR) of Sets with NULL $\space\emptyset$ which is nothing but the non-empty set itself. For example : $$\{q1 , q2 ,q0\}\cup\{\emptyset\} = \{q1 , q2 ,q0\}$$