Provide the correct solution for these, without using Chatgpt or copying from other sources:

$1.$ Show that a DFA Mk $=($Qk$,\backslash$Sigma k$,\backslash$delta k$,$ q$0,$ Fk$)$ exists that recognizes each Lk $=\{$w: w ends in k $0$s$\}$ over $\backslash$Sigma $=\{0,1\}.$

$($You will need to show a constructive proof as we did for $\{$w: w is a multiple of k$.\})$

$2.$ Let F be the language of all strings over $\{0,1\}$ that do not contain a pair of $1$s that are separated by an odd number of symbols. Give the state diagram of a DFA with five states that recognizes F$.$

$($You may find it helpful first to find a $4-$state NFA for the complement of F$.)$

$3.$ Show by giving an example that if M is an NFA that recognizes language C$,$ swapping the accept and nonaccept states in M doesn$$t necessarily yield a new NFA that recognizes the complement of C$.$ Is the class of languages recognized by NFAs closed under complement? Explain your answer.

$($What is the class of languages here?$)$