Give state diagrams of DFAs recognizing the following languages. In all parts sigma={0,1}.
i.{w| every odd position of w is 1}.
To create a state diagram for a DFA recognizing the language "every odd position of w is 1," we can follow the steps below:
1. Start with a single initial state.
2. Create two states: "Even" and "Odd."
3. Label the transition from the initial state to the "Even" state as "0."
4. Label the transition from the initial state to the "Odd" state as "1."
5. From the "Even" state, create a self-loop transition labeled "0."
6. From the "Even" state, create a transition to the "Odd" state labeled "1."
7. From the "Odd" state, create a self-loop transition labeled "1."
8. From the "Odd" state, create a transition to the "Even" state labeled "0."
The state diagram will look like this:
```
[ Initial State]
|
v
[ Even State]
| ^
| |
v |
[ Odd State ]
```
Now, let's label the transitions:
```
0
[ Initial State] -----> [Even State]
|
| 1
v
[ Odd State ]
```
This DFA will accept any input string where every odd position (counting from the left) is 1. For example, "1101," "1111," and "10101" would be accepted, while "0101," "1110," and "100110" would not be accepted.