The document presents proofs of logical equivalences using sentential logic. It provides proofs for the following equivalences: 1. (P ∧ Q) ≡ ¬(¬P ∨ ¬Q) 2. (P ∧ Q) ≡ ¬(P → ¬Q) 3. (P ∨ Q) ≡ ¬P → Q 4. P ≡ ((P ∧ Q) ∨ (P ∧ ¬Q)) 5. ¬(P → Q) ≡ (P ∧ ¬Q) Each proof uses rules of inference such as proof by contradiction (RAA) and

1. Disciplina de Estruturas L´ogico-Dedutivas
Solu¸c˜oes da Lista de Exerc´ıcios 5
M´etodo dedutivo da l´ogica sentencial: provas
de equivalˆencias
1. (P ∧ Q) ≡ ¬(¬P ∨ ¬Q)
1 P ∧ Q [H p/PC]
2 ¬P ∨ ¬Q [H p/RAA]
3 ¬P [H p/PC]
4 P ∧ Q [H p/RAA]
5 P [∧E 4]
6 ¬P ∧ P [∧I 3,5]
7 ¬(P ∧ Q) [RAA 4-6]
8 ¬P → ¬(P ∧ Q) [PC 3-7]
9 ¬Q [H p/PC]
10 P ∧ Q [H p/RAA]
11 Q [∧E 10]
12 ¬Q ∧ Q [∧I 9,11]
13 ¬(P ∧ Q) [RAA 10-12]
14 ¬Q → ¬(P ∧ Q) [PC 9-13]
15 ¬(P ∧ Q) [∨E 2,8,14]
16 ¬(P ∧ Q) ∧ (P ∧ Q) [∧I 15,1]
17 ¬(¬P ∨ ¬P) [RAA 2-16]
18 (P ∧ Q) → ¬(¬P ∨ ¬Q) [PC 1-17]
19 ¬(¬P ∨ ¬Q) [H p/PC]
20 ¬P [H p/RAA]
21 ¬P ∨ ¬Q [∨I 20]
22 ¬(¬P ∨ ¬Q) ∧ (¬P ∨ ¬Q) [∧I 19,21]
23 ¬¬P [RAA 20-22]
24 P [¬E 23]
25 ¬Q [H p/RAA]
26 ¬P ∨ ¬Q [∨I 25]
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2. 27 ¬(¬P ∨ ¬Q) ∧ (¬P ∨ ¬Q) [∧I 19,26]
28 ¬¬Q [RAA 25-27]
29 Q [¬E 28]
30 P ∧ Q [∧I 24,29]
31 ¬(¬P ∨ ¬Q) → (P ∧ Q) [PC 19-30]
32 (P ∧ Q) ≡ ¬(¬P ∨ ¬Q) [≡I 18,31]
2. (P ∧ Q) ≡ ¬(P → ¬Q)
1 P ∧ Q [H p/PC]
2 P → ¬Q [H p/RAA]
3 P [∧E 1]
4 Q [∧E 1]
5 ¬Q [MP 3,2]
6 Q ∧ ¬Q [∧I 4,5]
7 ¬(P → ¬Q) [RAA 2-6]
8 (P ∧ Q) → ¬(P → ¬Q) [PC 1-7]
9 ¬(P → ¬Q) [H p/PC]
10 ¬(P ∧ Q) [H p/RAA]
11 P [H p/PC]
12 Q [H p/RAA]
13 P ∧ Q [∧I 11,12]
14 ¬(P ∧ Q) ∧ (P ∧ Q) [∧I 10,13]
15 ¬Q [RAA 12-14]
16 P → ¬Q [PC 11-15]
17 ¬(P → ¬Q) ∧ (P → ¬Q) [∧I 9,16]
18 ¬¬(P ∧ Q) [RAA 10-17]
19 P ∧ Q [¬E 18]
20 ¬(P → ¬Q) → (P ∧ Q) [PC 9-19]
21 (P ∧ Q) ≡ ¬(P → ¬Q) [≡I 8,20]
3. (P ∨ Q) ≡ ¬P → Q
1 P ∨ Q [H p/PC]
2 ¬P [H p/PC]
3 P [H p/PC]
4 ¬Q [H p/RAA]
5 ¬P ∧ P [∧I 2,3]
6 ¬¬Q [RAA 4-5]
7 Q [¬E 6]
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3. 8 P → Q [PC 3-7]
9 Q [H p/PC]
10 Q → Q [PC 9-9]
11 Q [∨E 1,8,10]
12 ¬P → Q [PC 2-11]
13 (P ∨ Q) → (¬P → Q) [PC 1-12]
14 ¬P → Q [H p/PC]
15 ¬(P ∨ Q) [H p/RAA]
16 ¬P [H p/RAA]
17 Q [MP 16,14]
18 P ∨ Q [∨I 17]
19 ¬(P ∨ Q) ∧ (P ∨ Q) [∧I 15,18]
20 ¬¬P [RAA 16-19]
21 P [¬E 20]
22 P ∨ Q [∨I 21]
23 ¬(P ∨ Q) ∧ (P ∨ Q) [∧I 15,22]
24 ¬¬(P ∨ Q) [RAA 15-23]
25 P ∨ Q [¬E 24]
26 (¬P → Q) → (P ∨ Q) [PC 14-25]
27 (P ∨ Q) ≡ (¬P → Q) [≡I 13,26]
4. P ≡ ((P ∧ Q) ∨ (P ∧ ¬Q))
1 P [H p/PC]
2 ¬((P ∧ Q) ∨ (P ∧ ¬Q)) [H p/RAA]
3 ¬(P ∧ Q) [H p/RAA]
4 ¬Q [H p/RAA]
5 P ∧ ¬Q [∧I 1,4]
6 (P ∧ Q) ∨ (P ∧ ¬Q) [∨I 5]
7 ¬((P ∧ Q) ∨ (P ∧ ¬Q)) ∧ ((P ∧ Q) ∨ (P ∧ ¬Q)) [∧I 2,6]
8 ¬¬Q [RAA 4-7]
9 Q [¬E 8]
10 P ∧ Q [∧I 1,9]
11 ¬(P ∧ Q) ∧ (P ∧ Q) [∧I 3,10]
12 ¬¬(P ∧ Q) [RAA 3-11]
13 P ∧ Q [¬E 13]
14 (P ∧ Q) ∨ (P ∧ ¬Q) [∨I 13]
15 ¬((P ∧ Q) ∨ (P ∧ ¬Q)) ∧ ((P ∧ Q) ∨ (P ∧ ¬Q)) [∧I 2,14]
16 ¬¬((P ∧ Q) ∨ (P ∧ ¬Q)) [RAA 2-15]
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4. 17 (P ∧ Q) ∨ (P ∧ ¬Q) [¬E 16]
18 P → ((P ∧ Q) ∨ (P ∧ ¬Q)) [PC 1-17]
19 (P ∧ Q) ∨ (P ∧ ¬Q) [H p/PC]
20 P ∧ Q [H p/PC]
21 P [∧E 20]
22 P ∧ Q → P [PC 20-21]
23 P ∧ ¬Q [H p/PC]
24 P [∧E 23]
25 P ∧ ¬Q → P [PC 23-24]
26 P [∨E 19,22,25]
27 ((P ∧ Q) ∨ (P ∧ ¬Q)) → P [PC 19-26]
28 P ≡ ((P ∧ Q) ∨ (P ∧ ¬Q)) [≡I 18,27]
5. ¬(P → Q) ≡ (P ∧ ¬Q)
1 ¬(P → Q) [H p/PC]
2 ¬(P ∧ ¬Q) [H p/RAA]
3 P [H p/PC]
4 ¬Q [H p/RAA]
5 P ∧ ¬Q [∧I 3,4]
6 ¬(P ∧ ¬Q) ∧ (P ∧ ¬Q) [∧I 2,5]
7 ¬¬Q [RAA 4-6]
8 Q [¬E 7]
9 P → Q [PC 3-8]
10 ¬(P → Q) ∧ (P → Q) [∧I 1,9]
11 ¬¬(P ∧ ¬Q) [RAA 2-10]
12 P ∧ ¬Q [¬E 11]
13 ¬(P → Q) → (P ∧ ¬Q) [PC 1-12]
14 P ∧ ¬Q [H p/PC]
15 P → Q [H p/RAA]
16 P [∧E 14]
17 Q [MP 16,15]
18 ¬Q [∧E 14]
19 Q ∧ ¬Q [∧I 17,18]
20 ¬(P → Q) [RAA 15-19]
21 (P ∧ ¬Q) → ¬(P → Q) [PC 14-20]
22 ¬(P → Q) ≡ (P ∧ ¬Q) [≡I 13,21]
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