Problems:

$1\mathrm{.}2$ #$14$ a$,$ b$,$ e$,$ f Make sure each grammar is accompanied by an argument

Find grammars for $\backslash$Sigma $=\{$a$,$b$\}$ that generate the sets of

a$.$ all strings with exactly two a $\text{'}$s$.$

b$.$ all strings with at least two a$\text{'}$s$.$

c$.$ all strings with no more than three a$\text{'}$s$.$

d$.$ all strings with at least three a $\text{'}$s$.$

e$.$ all strings that start with a and end with b$.$

f$.$ all strings with an even number of b $\text{'}$s$.$

In each case, give convincing arguments that the grammar you give does indeed generate the indicated language.

#$20$

Find three strings in the language generated by

S$->$aSb$|$bSa$|$a

#$23$

Show that the grammars

S$->$aSb$|$ab$|\backslash$lambda

and

S$->$aaSbb$|$aSb$|$ab$|\backslash$lambda

are equivalent.