$2$ Syntax of First$-$order Logic

Draw the parse trees for the following first$-$order formulas and list the free and bound variables of

the formula. Furthermore, for each formula create a table with one row for each occurrence of each

atomic formula. The table should have the following columns.

The atomic formula.

The position of the occurrence of the atomic formula.

The free variables in the atomic formula at this occurrence.

The bound variables in the atomic formula at this occurrence.

As usual, $x,y,$ and $z$ are variables. Note that the same atomic formula can occur multiple time at

different positions.

$x=x^{{?}^{?}}$AAx$(P(x,y)$EEy$(Q(x,y)vvR(z)))$

EExP$(x,y{)}^{{?}^{?}}$AAyQ$(y,x)harrP(a,z)$