Find Proofs for the following 5 propositional logic statements

[josephmary]

i came acroos the below while studying propositional Logic, can anyone find the proofs

1) P ⊢ P

2) P → Q, Q→R ⊢ P → R

3) P → Q, Q→R, ¬R ⊢ ¬P

4) Q→R ⊢ (PvQ) → (PvR)

5) P →Q ⊢ (P&R) → (Q&R)

 

[HallsofIvy]

Can you state the axioms you are allowed to use?

 

[solakis1]

i came acroos the below while studying propositional Logic, can anyone find the proofs

1) P ⊢ P

2) P → Q, Q→R ⊢ P → R

3) P → Q, Q→R, ¬R ⊢ ¬P

4) Q→R ⊢ (PvQ) → (PvR)

5) P →Q ⊢ (P&R) → (Q&R)

1)$P$..................Assumption

2)$\neg P$................Hypothesis for contradiction

3)$P\wedge\neg P$............(1),(2) and using addition Introdaction

4)$\neg\neg P$..................From (2) to (3) and using contradiction

5) $P$..................(4) negation elimination

(2) and (3) are easy to do ,you can use hypothetical syllogism for (2) or conditional proof and modus ponens

And hypothetical syllogism , contrapositive and modus ponens for (3) or contradiction,and modus ponens

I will do (4) :

1)$Q\Rightarrow R$..............Assumption

2)$P\vee Q$..................Hypothesis for conditional proof

3)$\neg(P\vee R)$................Hypothesis for contraction

4)$(\neg P\wedge\neg R)$............From (3) and using de Morgan

5)$\neg P$..................(4), Addition elimination (AE)

6)$\neg R$..................(4),AE

7)$\neg R\Rightarrow\neg Q$.............(1),Contrapositive

8)$\neg Q$..................(6),(7),Modus Ponens(MP)

9)$\neg P\Rightarrow Q$..............(2),material implication

10)$Q$.....................(5),(9) MP

11)$Q\wedge\neg Q$................(8),(10) Addition Introduction (AI)

12)$\neg\neg(P\vee R)$...............from (3) to (11) and using contradiction

13)$(P\vee R)$...................(12),negation elimination

14)$(P\vee Q)\Rightarrow(P\vee R)$............from (2) to (13) and using conditional proof

(5) is on the same style with (4) and even easier