Each of the following language is the complement of a simpler language. In each part, construct a DFA for the simpler language, then use it to give the state diagram of a DFA for the language given. In all parts � sigma={a,b}

g. {w|w is any string that doesn't contain exactly two a's}
h.{w|w is any string except a and b}.

To construct a DFA for the simpler language, which is the complement of the given language, we need to follow a systematic approach.

Let's start with part (g) where the language is defined as {w | w is any string that doesn't contain exactly two a's}.

To construct a DFA for the simpler language, we need to create a DFA that accepts all strings that contain exactly two a's.

Step 1: Define the DFA state set
Create four states to represent different scenarios within the string:
- State 1: Initial state
- State 2: Contains one a
- State 3: Contains two a's (final/accepting state)
- State 4: Contains more than two a's

Step 2: Define the alphabet
The alphabet sigma is given as {a, b}.

Step 3: Define the transition function
Based on the alphabet, define the transition function for each state and input symbol.
- In State 1, if we get 'a', move to State 2. If we get 'b', stay in State 1.
- In State 2, if we get 'a', move to State 3. If we get 'b', move to State 4.
- In State 3, if we get any symbol (either 'a' or 'b'), stay in State 3 (final state).
- In State 4, if we get any symbol (either 'a' or 'b'), stay in State 4.

Step 4: Define the initial state
The initial state is State 1.

Step 5: Define the accepting/final states
The accepting state is State 3 since it represents exactly two a's.

Step 6: Construct the state diagram
Using the information from steps 1 to 5, construct the DFA state diagram for the simpler language.

Once you have constructed the DFA for the simpler language, we can use it to obtain the DFA for the given language by complementing the final/accepting states.

For part (g), the state diagram of the DFA for the language {w | w is any string that doesn't contain exactly two a's} would be the state diagram obtained from the DFA for the simpler language. However, the accepting state(s) would be complemented (if State 3 was the accepting state in the simpler language, it would not be accepting in the complement language, and vice versa).

Now, let's move to part (h) where the language is defined as {w | w is any string except 'a' and 'b'}.

To construct a DFA for the simpler language, we need to create a DFA that accepts all strings that only contain 'a' and 'b'.

Step 1: Define the DFA state set
Create three states:
- State 1: Initial state
- State 2: Contains 'a'
- State 3: Contains 'b' (final/accepting state)

Step 2: Define the alphabet
The alphabet sigma is given as {a, b}.

Step 3: Define the transition function
Based on the alphabet, define the transition function for each state and input symbol.
- In State 1, if we get 'a', move to State 2. If we get 'b', move to State 3.
- In State 2, if we get 'a', stay in State 2. If we get 'b', move to State 3.
- In State 3, if we get any symbol (either 'a' or 'b'), stay in State 3 (final state).

Step 4: Define the initial state
The initial state is State 1.

Step 5: Define the accepting/final states
The accepting state is State 3 since it represents only 'a' and 'b'.

Step 6: Construct the state diagram
Using the information from steps 1 to 5, construct the DFA state diagram for the simpler language.

Once you have constructed the DFA for the simpler language, we can use it to obtain the DFA for the given language by complementing the final/accepting state(s).

For part (h), the state diagram of the DFA for the language {w | w is any string except 'a' and 'b'} would be the state diagram obtained from the DFA for the simpler language. However, the accepting state(s) would be complemented (if State 3 was the accepting state in the simpler language, it would not be accepting in the complement language, and vice versa).

By following these steps, you will be able to construct DFAs for the simpler languages and then use them to obtain DFAs for the given languages.