$2\mathrm{.}57.$ Each case below defines a language over $\{a,b\}.$ In each case, decide whether the language can be accepted by an FA$,$ and prove that your answer is correct.

a$.$ The set of all strings $x$ beginning with a nonnull string of the form $ww.$

b$.$ The set of all strings $x$ containing some nonnull substring of the form $ww.$

c$.$ The set of all strings $x$ having some nonnull substring of the form $www.($You may assume the following fact: there are arbitrarily long strings in $\{a,b{\}}^{**}$ that do not contain any nonnull substring of the form w$$w$.)$

d$.$ The set of odd$-$length strings with middle symbol $a.$

e$.$ The set of even$-$length strings of length at least $2$ with the two middle symbols equal.

f$.$ The set of strings of the form $xyx$ for some $x$ with $|x|1.$

g$.$ The set of non$-$palindromes.

h$.$ The set of strings in which the number of $a\text{'}$s is a perfect square.

i$.$ The set of strings having the property that in every prefix, the number of $a\text{'}$s and the number of $b\text{'}$s differ by no more than $2.$