$2\mathrm{.}1.$ In each part below, draw an FA accepting the indicated language over $\{a,b\}.$

f$.$ The language of all strings in which the number of $a\text{'}$s is even.

$2\mathrm{.}2.$ For each of the FAs pictured in Fig. $2\mathrm{.}43,$ give a simple verbal description of the language it accepts.

$2\mathrm{.}3.$ a$.$ Draw a transition diagram for an FA that accepts the string abaa and no other strings.

b$.$ For a string xin$\{a,b{\}}^{**}$ with $|x|=n,$ how many states are required for an FA accepting $x$ and no other strings? For each of these states, describe the strings that cause the FA to be in that state.

c$.$ For a string xin$\{a,b{\}}^{**}$ with $|x|=n,$ how many states are required for an FA accepting the language of all strings in $\{a,b{\}}^{**}$ that begin with $x?$ For each of these states, describe the strings that cause the FA to be in that state.

$($a$)$

$($e$)$

Figure $2\mathrm{.}43|$