Matematica Discreta

Prob#01
Demostrar por reglas de inferencias que el razonamiento siempre es valido
∀x¬Px ∧Qx
∀y¬(Px V Q(x))
……………………………………………………..

∀¬Px V ¬Q(x) ∧ Vx¬Px ∧ ¬Q(x)

DE |∀x¬(P(x)∧Q(x)) ; | ∀x¬(P(x)VQ(x)) |   | } | ∀x[¬P(x)V¬Q(x)]∧∀x[¬P(x)∧¬Q(x)] |
1 | ∀x¬(P(x)∧Q(x)) | premisa 1 | | | | | |
2 | ∀x¬(P(x)VQ(x)) | premisa 2 | | | | | |
3 | ¬(P(x)∧Q(x)) | ∫_X^X ,1,UI | || | | |
4 | ¬(P(x)VQ(x)) | ∫_X^X ,2,UI | | | | | |
5 | DE | ¬[¬P(x)V¬Q(x)] | } | [P(x)∧Q(x)]∧¬[P(x)∧Q(x)] |   |
  | 5.1 | ¬[¬P(x)V¬Q(x)] | premisa 1 | | | | |
  | 5.2 | DE | ¬Q(x)| } | [¬P(x)V¬Q(x)]∧¬[¬P(x)V¬Q(x)] |
  |   | 5.2.1 | ¬Q(x) | | | premisa 1 | |
  |   | 5.2.2 | ¬P(x)V¬Q(x) | | | V-I,5.2.1 | |
  |   | 5.2.3 | [¬P(x)V¬Q(x)]∧¬[¬P(x)V¬Q(x)] | ∧-I,5.2.2,5.1| |
  | 5.3 | Q(x) | ¬-E,5.2 | | | | |
  | 5.4 | DE | ¬P(x) | } | [¬P(x)V¬Q(x)]∧¬[¬P(x)V¬Q(x)] |
  |   | 5.4.1 | ¬P(x) | | | premisa 1 | |
  |   | 5.4.2 | ¬P(x)V¬Q(x) | | |V-I,5.4.1 | |
  |   | 5.4.3 | [¬P(x)V¬Q(x)]∧¬[¬P(x)V¬Q(x)] | ∧-I,5.4.2,5.1 | |
  | 5.5 | P(x) | ¬-E,5.4 | | | | |
  | 5.6 | P(x)∧Q(x) | ∧-I,5.3,5.5 | | | | |
  | 5.7 |[P(x)∧Q(x)]∧¬[P(x)∧Q(x)] | | ∧-I,5.6,3 | | |
6 | [¬P(x)V¬Q(x)] | | ¬-E,5 | | | | |
7 | [P(x) → ¬Q(x)] | | def. -> ,6 | | | | |
8 | DE | ¬[¬P(x)∧¬Q(x)] | } | [P(x)VQ(x)]∧¬[P(x)VQ(x)] | |
  | 8.1 |¬[¬P(x)∧¬Q(x)] | premisa1 | | | | |
  | 8.2 | DE | ¬P(x) | } | ¬[¬P(x)∧¬Q(x)]∧[¬P(x)∧¬Q(x)] |
  |   | 8.2.1 | ¬P(x) | premisa 1 | | | |
  |   | 8.2.2 | P(x) | ¬-E,8.2.1 | | | |
  |   | 8.2.3 |¬Q(x) | →-E,8.2.2,7 | | | |
  |   | 8.2.4 | [¬P(x)∧ ¬Q(x)] | ∧-I,8.2.1,8.2.3 | |
  |   | 8.2.5 | ¬[¬P(x)∧¬Q(x)]∧[¬P(x)∧¬Q(x)] | ∧-I,8.2.4,8.1 | |
  | 8.3 | P(x) | ¬-E,8.2 | | | | |
  |8.4 | P(x)VQ(x) | V-I,8.3 | | | | |
  | 8.5 | [P(x)VQ(x)]∧¬[P(x)VQ(x)] | | ∧-I,8.4,4 | | |
9 | [¬P(x)∧¬Q(x)] | ¬-E,8 | | | | | |
10 | ∀x[¬P(x)V¬Q(x)] | GU,6 | |...