i dont understand what i did wrong here this is what my teacher said $1$ a$.$ b$.$ c$.$ expansions do not have quantifiers in them $2$ a$.$ b$.$ c$.$ names are not variables revise $1.$ Expand in a two$-$object universe $($the objects are named $\text{'}$a$\text{'}$ and $\text{'}$b$\text{'})$

$($a$)($x$)(($Ay v Bx$)$ Da$)$

$($Ax$)(($Aa v Ba$)$ Da$)($Bx$)(($Ab v Bb$)$ Da$)$

$($b$)$ ~ $($Ex$)$ ~ $($Ay $$ Cx$)$

~$($Ex$)(($Ay & ~Ca$)($Ay & ~Cb$))$

$($c$)($Ez$)$ ~ ~ $($Dz $($Hx & ~Gx$))$

$($Ez$)(($~Dz & ~$($Hz & ~Gz$)))$

$2.$ For the following wffs$,$ indicate which variables are free and which are bound $($you can use $\text{'}$F$\text{'}$ for free and $\text{'}$B$\text{'}$ for bound.$)$ Either $($i$)$ draw a vertical line underneath each variable with the letters $\text{'}$F$\text{'}$ or $\text{'}$B$\text{'}$ at the bottom of each vertical line or $($ii$)$ color bound variables red and free variables green. Make sure you know which symbols are used for variables. Not every symbol in the language of predicate logic is used for variables. The reading $($Scope$,$ binding, and quantifier expansions$)$ lists the symbols that are used for variables and the symbols that used for the names of objects $($i$.$e$.,$ individual constants$).$ Names of objects are not names of variables. The free$/$bound distinction only applies to variables. It does not apply to individual constants $($i$.$e$.,$ names of objects$)$

$($a$)($x$)($y$)($Ez$)(($Ayazwbd $$ Bzfhy$)$ v $($Fxzc $$ Hzzuju$))$

x is Bound by $($x$)$

y is Bound by $($y$)$

z is Bound by $($Ez$)$

w$,$ b$,$ d$,$ f$,$ h$,$ j$,$ u$,$ c All are free since they do not appear under any quantifier that binds them.

$($b$)($Ex$)($Ey$)($Hyabwa $($z$)($Fzvhky v Hxyyx$))$

x is Bound by $($Ex$)$

y is Bound by $($Ey$)$

z is Bound by $($z$)$

a$,$ b$,$ w$,$ v$,$ h$,$ k All are free since they do not appear under any quantifier that binds them.

$($c$)($z$)($Ex$)($y$)$ Axyzyyw $($y$)$ Bxyxzavz

$$ z is Bound by $($z$)$

$$ x is Bound by $($Ex$)$ in the antecedent, but free in the consequent as it is outside the scope of any quantifier binding x in that part.

$$ y Bound by the first $($y$)$ in the antecedent and by the second $($y$)$ in the consequent.

$$ a$,$ v All are free since they do not appear under any quantifier that binds them.