How I can explain this. Consider the following automaton, $A$.
Prove using the method of induction that every word/string $w\in L(A)$ contains an odd number(length) of $1$'s.
Show that there are words/strings with odd number(length) of $1$'s that does not belong to the language $L(A)$. Describe the language $L(A)$.
Here is what I did. For 1st question
q1 is the up-left circle , q2 the up-right, q3 down-left, q4 down-right.
and the transition table
0 | 1 -------------- q1 q3 q4 q2 q2 q1 q3 q3 q4 q4 q2 q1
basic inductive step: I verify that is valid for word = 1 (odd number of 1's) From state q1 we go to q4 (final accept state)
induction hypothesis: I assume that is valid for n = 2 * k +1 (n odd number 1's)
inductive step: 2(k+1) +1 I prove that is valid for 2(k+1) +1=> 2(k+1) +3=> 2(k+1)
For second Suppose the word =1000 or 10 with odd length of 1's , the final state is not the acceptance one.
Can anyone tell me if this I wrote is correct?