$2$A$.[6\%]$ Given the English sentence:

"Every student loves at least one course that some professor teaches."

Circle all correct logic translation$($s$)$:

A$.$x$($Student$($x$)$y$($Course$($y$)$ Loves$($x$,$ y$)$z$($Professor$($z$)$ Teaches$($z$,$ y$))))$

B$.$x$($Student$($x$)->$ Loves$($x$,$ y$))$y$($Course$($y$))$z$($Professor$($z$)$ Teaches$($z$,$ y$))$

C$.$x$($Student$($x$)->($y$($Course$($y$)$z$($Professor$($z$)$ Teaches$($z$,$ y$)$ Loves$($x$,$ y$)))))$

D$.$x$($Student$($x$)$y$($Course$($y$)->($z$($Professor$($z$)$ Teaches$($z$,$ y$))->$ Loves$($x$,$ y$))))$

E$.$x$($Student$($x$)$y$($Course$($y$)->$z$($Professor$($z$)$ Teaches$($z$,$ y$)$ Loves$($x$,$ y$))))$

F$.($Student$($x$)$ Professor$($z$)$ Teaches$($z$,$ y$))($Student$($x$)$ Course$($y$)$ Teaches$($z$,$ y$))$

$2$B$.[6\%]$ Given the English sentence:

"For any project, the project is delayed if and only if at least one of its tasks is

not completed on time."

Circle all correct logic translation$($s$)$:

A$.$x$($Project$($x$)->$ Delayed$($x$)->$y$($Task$($y$)$ PartOf$($y$,$ x$)$CompletedOnTime$($y$)))$

B$.$x$($Project$($x$)$ Delayed$($x$))->($y$($Task$($y$)$ PartOf$($y$,$ x$)->$CompletedOnTime$($y$)))$

C$.$x$($Project$($x$)($y$($Task$($y$)$ PartOf$($y$,$ x$)->$ CompletedOnTime$($y$))))->$Delayed$($x$)$

D$.$x$($Project$($x$)$ Delayed$($x$))($y$($Task$($y$)$ PartOf$($y$,$ x$)->$ CompletedOnTime$($y$)))$

E$.$x$($Project$($x$)->($Delayed$($x$)$y$($Task$($y$)$ PartOf$($y$,$ x$)$CompletedOnTime$($y$))))$

F$.$x$(($Project$($x$)$Delayed$($x$)($y$($Task$($y$)$ PartOf$($y$,$ x$)$CompletedOnTime$($y$)))($Project$($x$)$

Delayed$($x$)($y$($Task$($y$)$ PartOf$($y$,$ x$)$CompletedOnTime$($y$))))$

$2$C$.[8\%]$ Transform the following

logic sentence to CNF$.$

NOTE:

$$ Use lower$-$case letters such as $$x$,$y$,$z$$ for variables.

$$ Use upper$-$case letters such as $$A$,$B$,$C$$ for constants.

$$ Use lower$-$case letters such as $$g$($x$),$h$($x$)$ for skolem functions, if needed.

$$ Do NOT use confusing notations when answering the question. For example, you should not use lower

case $$x$$ for a variable and upper case $"$X$$ for a constant at the same time.

$("$x$)($P$($x$)=>(("$y$)($P$($y$)=>$ P$($x$,$y$))($$y$)($Q$($x$,$y$)$ P$($y$))))$

Eliminate $=>$

Reduce scope of negation

Standardize variables

Eliminate existential quantification

Drop universal quantification symbols

Convert to conjunction of disjunctions

Create separate clauses

Standardize variables