Problem $23.$ Prove rigorously $($by induction on an appropriate quantity$)$ that each of the following grammars

generates the language $\{\begin{array}{ccc}\text{x}\text{i}\text{n}& \{0,& 1{\}}^{**}|\text{\#}0\end{array}\text{'}$s in $x=$#$1\text{'}$s in $\{$:$x\}.$

$($Homework $3$ Problem $3)V=\{E\}$ with production rules $E0E1E|1E0E|.$

$V=\{E\}$ with production rules $E0E1|1E0|EE|.$

Problem $24($Homework $3$ Problem $4).$ Prove rigorously $($by induction on an appropriate quantity$)$ that

the grammar with a single non$-$terminal $E$ and production rules $E0E1E,E$ generates the language

$\{xin\{0,1{\}}^{**}|$#$0\text{'}sinx=$#$1\text{'}sinx,$ and for every prefix $x^{\text{'}}ofx,$#$0\text{'}sin{x}^{\text{'}}$#$1\text{'}sin{x}^{\text{'}}\}.$

Also prove that the grammar is unambiguous.

Problem $25.$ Design an unambiguous grammar that generates the language $\{\begin{array}{ccc}\text{x}\text{i}\text{n}& \{0,& 1{\}}^{**}|\text{\#}0\end{array}\text{'}$s in $x=$

#$1\text{'}$s in $x.($Hint: take help from Problem $23$ and Problem $24.)$

Problem $26.$ Determine with proof the language generated by the grammar $G=(\{S\},\{a,b\},P,S),$ where

the set of production rules is

$P=\{SlongrightarrowSS,$SlongrightarrowaaSb,SlongrightarrowbSaa,SlongrightarrowaSbSa,Slongrightarrow$\}.$