Constructing DFAs $(30$ points; each part $10$ points$)$

For each of the following languages over the alphabet $\backslash$Sigma $=\{0,1\},$ give a DFA that recognizes the language,

and explain why it$$s correct. Try to use as few states as possible.

$($a$)\{$w in $\backslash$Sigma $|$ w has an odd number of $1$s$\}.$ For instance, $11111$ has five $1$s and is in the language, and

$001000$ has one $1$ and is in the language, while $11001010$ has four $1$s and is not in the language.

$($b$)\{$w in $\backslash$Sigma $|$ w starts and ends with the same character$\}.$ For instance, $100101$ starts and ends with $1$

and is in the language, and $001000$ starts and ends with $0$ and is in the language, while $11001010$

starts with $1$ but ends with $0$ and is not in the language.

$($c$)\{$w in $\backslash$Sigma $|$ w starts and ends with the same character and also has an odd number of $1$s$\}.$