Solución

a.) (𝑝 ∨ 𝑞) →∼ 𝑟

b.) (𝑝 ∧ 𝑞) → ∼ (𝑝 ∧ 𝑟)

c.) ∼ (p →∼ 𝑟)

d.) ∼ (𝑟 ∨ 𝑞) →∼ 𝑝

Solución

a).

↔

∼

→

𝑟 p

→

⊻

∧

∼

𝑝

𝑞 ∼

𝑝

𝑝

b).

p r s ~P ~p↔r (pVs) [(~p<->r) ← (p V s)] ~r (p⊻~r) [(~p<->r) <- (p V s)] ⌅(p⊻~r)

F F F V F F F V F V

F F `` V F V V V F V

F V F V V F V F V F

F V V V V V V F V F

V F F F V V V V F V

V F V F V V V V F V

V V F F F V V F F V

V V V F F V V F F V

p q r ~P (~p ^ q) ~r (p V ~r) [(~p ^ q) → (p V ~r)] (p → r) ~ (p → r) [(~p ^ q) → (p V ~r)] ↔ ~ (p → r)

F F F V F V V V V F F

F F V V F F F V V F F

F V F V V V V V V F F

F V V V V F F F V F V

V F F F F V V V F V V

V F V F F F V V V F F

V V F F F V V V F V V

V V V F F F V V V F F

s

⊻

~

𝑟 p

←

V

⇿

𝑝

𝑟 ∼

𝑝

⌅

c).

p r s ~p (~p↔r) ~s (p ∨ ~s) [(~p↔r) ↚ (p ∨ ~s)] ~r (~p∧~r) ~(~p∧~r) [(~p↔r) <-/-

(p ∨ ~s)]∧ ~(~p∧~r)

F F F V F V V V V V F F

F F V V F F F F V V F F

F V F V V V V F F F V F

F V V V V F F F F F V F

V F F F V V V F V F V F

V F V F V F V F V F V F

V V F F F V V V F F V V

V V V F F F V V F F V V

Solución

a).

p q r ~p (~p→q) (p∨r) (~p→q) ↚ (p∨r)

F F F V F F F

F F V V F V V

F V F V V F F

F V V V V V F

⊻

~

𝑟 p

↚

V

∧

𝑝

𝑟 ∼

𝑝

∧

~

∼

s

V F F F V V F

V F V F V V F

V V F F V V F

V V V F V V F

𝐹𝑁𝐶 = (𝑝𝑉𝑞𝑉𝑟) ∧ (𝑝𝑉~𝑞𝑉𝑟) ∧ (𝑝𝑉~𝑞𝑉~𝑟) ∧ (~𝑝𝑉𝑞𝑉𝑟) ∧ (~𝑝𝑉𝑞𝑉~𝑟) ∧ (~𝑝𝑉~𝑞𝑉𝑟)

∧ (~𝑝𝑉~𝑞𝑉~𝑟)

𝐹𝑁𝐷 = (~𝑝 ∧ ~𝑞 ∧ 𝑟)

b).

p q r s ~p (~p←r) (p∨s) [(~p←r)→(p∨s)] (p∧q) ~(p∧q) [(~p←r)→(p∨s)] ↮ ~(p∧q)

F F F F V V F F F V V

F F F V V V V V F V F

F F V F V V F F F V V

F F V V V V V V F V F

F V F F V V F F F V V

F V F V V V V V F V F

F V V F V V F F F V V

F V V V V V V V F V F

V F F F F V V V F V F

V F F V F V V V F V F

V F V F F F V V F V F

V F V V F F V V F V F

V V F F F V V V V F V

V V F V F V V V V F V

V V V F F F V V V F V

V V V V F F V V V F V

𝐹𝑁𝐶 = (𝑝 𝑉 𝑞 𝑉 𝑟 𝑉 ~𝑠)˄(𝑝 𝑉 𝑞 𝑉 ~𝑟 𝑉 ~𝑠)˄(𝑝 𝑉 ~𝑞 𝑉 𝑟 𝑉 ~𝑠)˄(𝑝 𝑉 ~𝑞 𝑉 ~𝑟 𝑉 ~𝑠)˄ (~𝑝 𝑉 𝑞 𝑉 𝑟 𝑉 𝑠)˄(~𝑝 𝑉 𝑞 𝑉 𝑟 𝑉 ~𝑠)˄(~𝑝 𝑉 𝑞 𝑉 ~𝑟 𝑉 𝑠)˄(~𝑝 𝑉 𝑞 𝑉 ~𝑟 𝑉 ~𝑠) 𝐹𝑁𝐷

= (~𝑝 ~𝑞 ~𝑟 ~𝑠)˅(~𝑝 ~𝑞 𝑟 ~𝑠)˅(~𝑝 𝑞 ~𝑟 ~𝑠)˅(~𝑝 𝑞 𝑟 ~𝑠)(𝑝 𝑞 ~𝑟 ~𝑠)

˅(𝑝 𝑞 ~𝑟 𝑠)˅(𝑝 𝑞 𝑟 ~𝑠)˅(𝑝 𝑞 𝑟 𝑠)

c).

p r s ~p (~p⟷r) (p⊻ s) [(~p⟷r)⟷(p⊻ s)] ~r (~p⟷~r) [(~p⟷r)⟷(p⊻ s)] ⌅ (~p⟷~r)

F F F V F V F V V V

F F V V F F V V V F

F V F V V V V F F V

F V V V V F F F F V

V F F F V F F V F V

V F V F V F F V F V

V V F F F F V F V F

V V V F F F V F V F

𝐹𝑁𝐶 = (𝑝𝑉𝑟𝑉~𝑠)˄(~𝑝𝑉~𝑟𝑉𝑠)˄(~𝑝𝑉~𝑟𝑉~𝑠) 𝐹𝑁𝐷 = (~𝑝^~𝑞^~𝑟)˅(~𝑝^𝑞^~𝑟)˅(~𝑝^𝑞^𝑟)˅(𝑝^~𝑞^~𝑟)˅(𝑝^~𝑞^𝑟)

𝑆1 = 𝐹𝑁𝐷 = (~𝑝^𝑞^~𝑟) ˅(𝑝^~𝑞^𝑟) 𝑆2 = 𝐹𝑁𝐷 = (~𝑝^~𝑞^~𝑟) ˅ (~𝑝^𝑞^~𝑟) ˅ (𝑝^~𝑞^~𝑟) ˅ (𝑝^~𝑞^𝑟) = ~𝑝 ˅ (~𝑞^~𝑟) ˅ (𝑞 ^~𝑟) ˅(𝑝^~𝑞) ˅(~𝑟^𝑟) = ~𝑝 [(~𝑞 ^ 𝑞) ^~𝑟] ˅(𝑝^~𝑞) ˅𝐹 = ~𝑝 ˅ 𝐹 ^~𝑟 ˅(𝑝^~𝑞) ˅ 𝐹 = (~𝑝 ˅~𝑟) ˅(𝑝 ^~𝑞) = ~(𝑝^𝑟) ˅(𝑝 ↛ 𝑞)

𝑆3 = 𝐹𝑁𝐶 = (𝑝 ˅𝑞 ˅𝑟)^(~𝑝 ˅ 𝑞˅𝑟)^(~𝑝 ˅~𝑞 ˅𝑟) = [(𝑞˅𝑟) ˅(~𝑝^𝑝)] ^(~𝑝 ˅~𝑞 ˅𝑟)

= [(𝑞˅𝑟) ˅𝐹] ^(~𝑝 ˅~𝑞 ˅𝑟) = (𝑞˅𝑟)^ (~𝑝 ˅~𝑞 ˅𝑟) = 𝑟˅[~𝑝˅ (𝑞^~𝑞)] = 𝑟˅[~𝑝˅ 𝑉] = 𝑟˅~𝑞

= 𝑟 ⟵ 𝑞 𝑆4 = 𝐹𝑁𝐷 = (~𝑝^~𝑞^~𝑟) ˅ (~𝑝^~𝑞^𝑟) ˅ (~𝑝^𝑞^~𝑟) = [(~𝑝^~𝑞) ˅(𝑟 ˅~𝑟)] ˅(~𝑝^𝑞^~𝑟) = [(~𝑝^~𝑞) ˅𝐹] ˅(~𝑝^𝑞^~𝑟) = (~𝑝^~𝑞) ˅(~𝑝^𝑞^~𝑟) = ~𝑝 ^(~𝑞 ˅𝑞) ˅~𝑟 = ~𝑝^𝐹 ˅~𝑟

= ~𝑝 ˅~𝑟 = ~(𝑝^𝑞)

1. p⟶~r

p ______________ ∴ ~r

2. p ⟶ ~ (q v s)

p ______________ ∴ ~( q v s)

3. p⟶~r

^~r ______________ ∴ P