Symbolic Logic Proofs?
Any help is greatly appreciated thanks. I need help finishing these proofs using any of the 18 implication and explication rules (MT,MP,DS,HS,Simp, Conj, Add, CD, Double Negation,DeMorgan's,Commutation,Associat… Contraposition, Implication, Exportation, Tautology, and Equivalence)
Proof 1
1. If P then [(Q and R) or S]
2. If (Q and R) then Not P
3.If T then Not S
...
Therefore, If P then Not T
Proof 2
1. If A then B
2. A or (B and C)
...
Therefore, B
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Answers & Comments
Verified answer
★ “ if p then q” is equivalent to “(not p) or q”
I think the statement below is obviously true.
☆ “ if p then q” and “ if p´ then q´” ⇒ if（p or p´）then (q or q´)
Proof 1
(1) If P then [(Q and R) or S]
(2) If (Q and R) then Not P
(3) If T then Not S
...
Therefore, If P then Not T
【Solution】
From （3）, T→not S ⇔ not (not S) → not T ⇔ S → not T ….(4)
From (2) , (Q and R) → Not P ….(5)
From (4), (5) and ☆ , (Q and R) or S → (Not P) or (not T) ….(6)
From (1) ,(6) and ★, P → (Not P) or (not T) ⇔ (not P) or (not P) or (not T)
⇔ (not P) or (not T) ⇔ P → (not T) ⇔ If P then Not T
Proof 2
(1) If A then B
(2) A or (B and C)
...
Therefore, B
【Solution】
From (2) and ★, (not A)→(B and C) ⇒ (not A) → B ….(3)
From (1) and (3) , { A or (not A) } → Ｂ
A or (not A) is always true.
Therefore B