$(15$ marks$)$ Design PDA and draw a diagram for each of the following:

$1.L=\{0^n{1}^m|nm\}$

$2.L=\{0^n{1}^m{0}^k|n+k=m\}$

$3.\{a^j{b}^j|0ij\}$

$4.\{a^i{c}^j{b}^i|i,j0\}$

$5.\{\begin{array}{c}{a}^i{b}^j{c}^k|i=j\end{array}$ or $\{$:$j=k\}$