QUESTION $2$

Question $2\mathrm{.}1$

Consider the following predicate symbols and their intended meanings:

$($i$)$ Express the following predicate logic formula in English, where the symbols have the meanings

as given above:

EEy$(P(y)J(y))$

$($i$)$ Express the following declarative sentence in predicate logic using the symbols as given above:

There is an advanced course such that every programmer has a final exam in it$.$

Question $2\mathrm{.}2$

Let

$P$ be a predicate symbol with one argument and $Q$ a predicate with one argument, respectively

$x,y$ are variables

State which of the following are well formed formulas:

$($i$)P(x{)}^{{?}^{?}}Q(x)$

$($ii$)P(x)$

$($iii$)$ EEx

$($iv$)$

Question $2\mathrm{.}3$

Consider the following formula $$ where $P,Q$ and $R$ are predicate symbols with one argument $.$:

$(AAx(x{)}^{{?}^{?}}$EEynotQ$(y))vvEEzR(z)$

$($i$)$ Draw the parse tree of $.$

$($ii$)$ Mark the free and bound variables on the tree.

$($iii$)$ Let $g$ be a function with one argument. Is $g(x)$ free for $y$ in $?$ Explain your answer.

Question $2\mathrm{.}4$

Using the basic natural deduction rules for predicate logic, prove the validity of the following sequent:

$($i$),$AAx$(P(x{)}^{{?}^{?}}Q(x)),$EEyS$(y)|--$EEz$(P(z{)}^{{?}^{?}}S(z))$

$($ii$)$ AAx$(P(x)(Q(x)vvR(x))),$notEEx$(P(x{)}^{{?}^{?}}R(x))|--$AAx$(P(x)Q(x))$