Logical reasoning validity using natural deduction

Question

For each of the following use natural deduction to provide a derivation which shows the argument to be valid.

1. ~P

~P → Q

Q

2. ~F

F ∨ ~G

~G ∨ H

3. ~X → Y

~X

Y → ~Z

~Z

Answer 1

1 ~P = Socrates is not a woman. Q= Socrates is a man. If Socrates is not a woman then Socrates is a man. Therefore, Socrates is a man.

2 ~F = Rittenhouse did not commit murder ~G = Rittenhouse is Not Guilty. Either Rittenhouse committed murder or he is Not Guilty H= Aliens landed in Hamburg Germany F v ~G, Therefore ~G or H

Since Rittenhouse did not commit murder, he must be Not Guilty. If he's not Guilty, then it's also true that he's either Not Guilty or anything else is true such as Aliens landing. H could mean anything.

3 ~X = No voter fraud in 2020 Y= Biden is president. ~Z = Trump is not president

if ~X then Y, if Y then ~Z, therefore ~Z Since there was no voter fraud, Biden is present. IF Biden is president, Trump is not president.