This question is about the pumping lemma for context$-$free languages.

$($a$)$ Let A $=\{$a

$2$

n

$|$ n $>=0\}.$

Suppose you are trying to prove that A is not context$-$free using the pumping

lemma for context$-$free languages. Your proof starts $($correctly$)$ like this:

Suppose for a contradiction that A is context$-$free. Let p be the pumping length

given by the pumping lemma for context$-$free languages.

Now you have to choose string s$.$ Give a good choice for s$.$ You do not have to

explain your choice and you do not have to finish the proof. Simply give s$.$

$($b$)$ Let B $=\{$ww $|$ w in $\{$a$,$ b$\}$

$\}.$

Suppose you are trying to prove that B is not context$-$free using the pumping

lemma for context$-$free languages. Your proof starts $($correctly$)$ like this:

Suppose for a contradiction that B is context$-$free. Let p be the pumping length

given by the pumping lemma for context$-$free languages.

Now you have to choose string s$.$ Give a good choice for s$.$ You do not have to

explain your choice and you do not have to finish the proof. Simply give s$.$

$($c$)$ Let C $=\{$w in $\{$a$,$ b$,$ c$,$ d$\}$

$$

$|$ w has the same number of a$$s as b$$s and w has the

same number of c$$s as d$$s$\}.$

Suppose you are trying to prove that C is not context$-$free using the pumping

lemma for context$-$free languages. Your proof starts $($correctly$)$ like this:

Suppose for a contradiction that C is context$-$free. Let p be the pumping length

given by the pumping lemma for context$-$free languages.

Now you have to choose string s$.$ Give a good choice for s$.$ You do not have to

explain your choice and you do not have to finish the proof. Simply give s