Showing that a Language is regular using a state machine diagram

Question

I'm in my first few weeks of taking a theoretical course at my school and was wondering what is wrong with my answer to this question.

I've been told to show that the language:

L = {x | x has even length and ends with b} over the alphabet {a,b}

Is regular

I know I can prove this by showing a DFA that accepts that language, over that alphabet. I came up with this solution:

(the letters are in color blocks because the background is black and with a transparent image they wouldn't show up on a black background).

https://i.stack.imgur.com/YbqVN.jpg

This seems to work on strings I've tested such as: ab

abbb

ababab

bbaabbababab

etc

However, my textbook provides a solution with 4 states instead of 3, so I'm wondering where I have gone wrong here? Is there a way to easily check what strings wouldn't work for this DFA? It seems trivial to test every string out there, as it would be impossible.

Answer 1

Your answer is correct. If the exercise don’t ask the minimal DFA then the response is not unique (you can make well states).

Answer 2

Note that it is not necessary to use automata to prove that your language is regular. Setting $A = \{a,b\}$, the language of words of even length is $(A^2)^*$ and the language of words ending with $b$ is $A^*b$. These two languages are regular and so is their intersection, which is your language $L$.

Now, the 3-state DFA you have computed turns out to be the minimal DFA of $L$, and hence your answer is correct.