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# State the worst case number of states in DFA and NFA for a language?

A Deterministic Finite automata (DFA) is a five tuples

M=(Q, ∑, δ,q0,F)

Where,

Q − Finite set called states.

∑ − Finite set called alphabets.

δ − Q × ∑ → Q is the transition function.

q0 ∈ Q is the start or initial state.

F − Final or accept state.

Let’s see the worst case number of states in DFA for the language **A∩B and A***

Let A and B be the two states,

|A| = number of states = nA

|B| = number of states = nB

DFA = |A∩B|

=nA.nB

|A ∪ B| =nA.nB

|A*|=3/4 2^{nA}

|AB| = nA (2^{nB}-2^{nB-1})

**NFA**

The non-deterministic finite automata (NFA) also have five states same as DFA, but with different transition function, as shown follows −

δ: Q X ∑ -> 2^{Q}

Where,

Q − Finite set of states.

∑ − Finite set of the input symbol.

q0 − Initial state.

F − Final state.

δ − Transition function.

Let’s see the worst case number of states in NFA for the language **A∩B and A***

Let A and B be the two states,

|A| = number of state = nA

|B|= number of state = nB

NFA:

|AUB| = nA+nB+1

|A*| = nA+1

|AB| = nA+nB

|A∩B| = nA.nB

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