QUESTION $4$

$10$

Let $$ be the formula

EEx$(notP(x,y)Q(x,y,z))$AAyEEzQ$(x,x,x{)}^{{?}^{?}}P(x,y)$

where $P$ is a predicate symbol with two arguments and $Q$ is a predicate symbol with three arguments.

Question $4\mathrm{.}1$

Draw the parse tree of the formula and indicate the free and bound variables.

Question $4\mathrm{.}2$

$(6)$

Suppose $f$ is a function symbol with one argument. For each of the following substitutions, state whether it will create a problem. If there is no problem, write down the substituted formula. If there will be a problem, state how you would solve it and then write down the substituted formula.

Question $4\mathrm{.}2\mathrm{.}1[f\frac{z}{x}]$

Question $4\mathrm{.}2\mathrm{.}2[f\frac{z}{y}]$

Question $4\mathrm{.}2\mathrm{.}3[f\frac{x}{y}]$