QUESTION $4$

$[15]$

$4\mathrm{.}1$ Transform the following formula into the prenex normal form:

AAx$((C(x{)}^{{?}^{?}}$EEy$(T(y{)}^{{?}^{?}}L(x,y)))$EEy$(D(y{)}^{{?}^{?}}B(x,y)))$

$4\mathrm{.}2$ Find the prenex normal form of AAx$(EEyR(x,y{)}^{{?}^{?}}$AAynotS$(x,y)$not$(EEyR(x,y{)}^{{?}^{?}}P))$

$4\mathrm{.}3$ Transform the following formula into prenex normal form:

AAx$(P(x)((AAy)(P(y)P(f(x,y)){)}^{{?}^{?}}$not$(AAy)(Q(x,y)P(y))))$