Logic Reasoning

Problem $1$: CNF Transformation on propositional logic

Given the following logical statements, convert them into Conjunctive Normal Form $($CNF$)$:

$$

$($P $$ Q$)$ R

$$

$$

$($P $$ Q$)($R $$ S$)$

$$

$$

$($P $$ Q$)($R $$ S$)$

$$

Problem $2$: Resolution on Propositional Logic

Apply the resolution rule to prove or disprove the following statement:

$$

$[($P $$ Q$)($P $$ R$)($R $$ S$)]$ S

$$

Problem $3$: First$-$Order Logic

Translate the following English sentences into first$-$order logic:

$1.$ All cats are mammals.

$2.$ Some birds can fly.

$3.$ Every person has a mother.

Problem $4$: CNF transformation on FOL

Given the following logical statement, convert them into Conjunctive Normal Form $($CNF$)$:

$$

$$x $($Cat$($x$)$y $($Mother$($y$,$ x$)))$z $($Mammal$($z$))$

$$

Problem $5$: Resolution on FOL

Here is the problem:

"Every student who studies computer science is enrolled in a programming course."$\backslash$

"All students who pass the math exam are eligible for the scholarship."$\backslash$

"John studies computer science."$\backslash$

$"$If John is eligible for the scholarship, then he passed the math exam."

Based on the given statements, determine whether John is eligible for the scholarship.

Please help me with this; just follow the instructions carefully. Thank you!!