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Ejercicios predicados
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Alfonso Cabanzo
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Ejercicios predicados
1.
Ejercicios de lógica de primer Orden Alfonso Cabanzo I. Realizar
las siguientes deducciones usando reglas básicas o derivadas de deducción natural para lógica de proposiciones, pero usando sólo reglas primitivas de lógica de predicados (,∀ܧ ,∀ܫ ,∃ܧ :)∃ܫ 1) –1. ∀ݔ൫ܲݔ ∧ ሺܴݔ ∧ ܵݔሻ൯ –2. ∀ݔܶݔ ⊢ ܵܽ ∧ ܶܽ 2) –1. ∀ݔሺܳݔ ∧ ܴݔሻ –2. ∀ݔሺܵݔ ∧ ܶݔሻ ⊢ ܴܽ ∧ ܾܵ 3) –1. ∀ݔሺܲݔ ∧ ܳݔሻ –2. ∀ݔሺܴݔ ∧ ܵݔሻ ⊢ ܳܽ ∧ ܾܵ 4) –1. ∀ݕݔሺܲݕݔ ∧ ܳݔሻ ⊢ ܲܽܽ 5) –1. ∀ݔܴݔ ∧ ∀ݕሺܶܽݕ → ܵܽݕሻ ⊢ ܾܶܽ → ܾܵܽ 6) –1. ∀ݔሺܲݔ ∧ ܴݔሻ ∧ ∀ݕܵݕ –2. ∀ݔሺݔܯ ∧ ܶݔሻ ∧ ∀ݔܳݔ ⊢ ∃ݔሺܲݔ ∧ ݔܯሻ ∧ ∃ݕሺܴݕ ∧ ܳݕሻ 7) –1. ∀ݕݔሺܲݔ ∧ ݕܯሻ ⊢ ∃ݔሺݔܯ ∧ ܲݔሻ 8) –1. ∀ݕݔ൫ሺܲݔ ∧ ܶݔሻ ∧ ܵݕ൯ ⊢ ∃ݔሺܵݔ ∧ ܶݔሻ ∨ ∀ݕݔܲݕ∃ݔ 9) –1. ∀ݕݔሺܲݔ ∧ ܶݕሻ –2. ∀ݔܯݔ ⊢ ∃ݔܲݔ ∧ ∀ݔܯݔ 10) –1. ∀ݔ൫ሺݔܯ → ܶݔሻ ∧ ሺܴݔ ↔ ܳݔሻ൯ –2. ∀ݔ൫ሺܵݔ ∨ ܼݔሻ ∧ ሺݔܤ → ܶݔሻ൯ ⊢ ∃ݓݖݕݔ൫ሺݔܯ → ܶݕሻ ∧ ሺݖܤ → ܶݓሻ൯
2.
Alfonso Cabanzo 2 11) –1. ∀ݔ൫ሺݔܯ
∨ ܵݔሻ ∧ ሺܶݔ ∨ ܹݔሻ൯ –2. ∀ݔ൫ሺܲݔ → ܳݔሻ ∧ ሺܴݔ → ܶݔሻ൯ ⊢ ∃ݕݔሺݔܯ ∨ ܵݕሻ ∧ ∃ݕሺܴݕ → ܶݕሻ 12) –1. ∀ݔ ሺݔܯ ∨ ܵݔሻ ∧ ∀ݕሺܶݕ → ܳݕሻ –2. ∀ݔ൫ܵݔ ∨ ሺܲݔ → ܶݔሻ൯ ⊢ ∃ݔሺܶݔ → ܳݔሻ ∨ ∃ݔ൫ܵݔ ∨ ሺܲݔ → ܶݔሻ൯ 13) –1. ∀ݔ ቀ൫ሺܲݔ → ܴݔሻ → ܳݔ൯ ∧ ൫ሺܲݔ → ܱݔሻ ↔ ሺܰݔ ∨ ݔܯሻ൯ቁ ⊢ ∃ݕݔ൫ሺܲݔ → ܴݔሻ → ܳݕ൯ 14) –1. ∀ݔ൫ሺݔܯ ∧ ܶݔሻ ∧ ሺݔܣ → ݔܤሻ൯ –2. ∀ݔሺሺܲݔ → ܳݔሻ ∧ ܵݔሻ ⊢ ∃ݖݔ൫ሺݔܯ ∧ ܶݔሻ ∧ ሺܲݖ → ܳݖሻ൯ 15) –1. ∀ݔሺܲݔ ∧ ܳݔሻ ⊢ ∀ݔሺܲݔ ∨ ܴݔሻ 16) –1. ∀ݔሺܵݔ ∧ ܶݔሻ ⊢ ∀ݔሺܵݔ ∨ ܶݔሻ 17) –1. ∀ݔ൫ሺܲݔ ∧ ܳݔሻ ∨ ܴݔ൯ ⊢ ∀ݔܲݔ ∨ ∀ݕܴݕ 18) –1. ∀ݔ൫ሺܲݔ → ܳݔሻ ∧ ሺܴݔ → ܵݔሻ൯ ⊢ ∀ݔሺܲݔ → ܳݔሻ ∨ ∃ݔሺܴݔ → ܵݔሻ 19) –1. ∀ݕݔܲݕݔ ⊢ ∀ݕݔܲݕ∃ݔ ∨ ∀ݔ൫ܴݔ → ሺܵݔ → ܶݔሻ൯ 20) –1. ∀ݔ൫ሺܲݔ ∨ ܴݔ → ܶݔሻ ∧ ሺܶݔ → ܵݔሻ൯ ⊢ ∀ݔሺܶݔ → ܵݔሻ ∨ ∃ݕݔሺܲݕݔ ∧ ܴݔݕ ↔ ܶݕݔሻ 21) –1. ∀ݔ൫ሺݔܣ ∧ ݔܤሻ ∧ ሺݔܥ ∧ ݔܦሻ൯ –2. ∃ݔ ൫ሺݔܧ → ݔܨሻ ∧ ሺݔܩ ↔ ݔܪሻ൯ ⊢ ∃ݔ൫ሺݔܣ ∧ ݔܥሻ ∧ ሺݔܧ → ݔܨሻ൯ ∨ ∀ݔሺݔܭ ∨ ܴݔሻ 22) –1. ∀ݔ ൫ሺݔܫ → ݔܬሻ ∨ ሺݔܭ → ݔܮሻ൯ ∨ ∃ݔ൫ሺݔܯ ∨ ܵݔሻ ∧ ሺܶݔ ∨ ܹݔሻ൯ ⊢ ∃ݔ൫ሺݔܭ → ݔܮሻ ∨ ሺܶݔ ∨ ܹݔሻ൯ ∨ ∀ݔ∃ݕሺݕܩ ∧ ݔܨ → ݔܣሻ
3.
Alfonso Cabanzo 3 23) –1. ∃ݔ൫ሺܲݔ ∨
ܳݔሻ ∨ ܴݔ൯ –2. ∀ݔሺܵݔ → ܶݔሻ ⊢ ∃ݔ ቀ൫ሺܲݔ ∨ ܳݔሻ ∨ ܴݔ൯ ∧ ሺܵݔ → ܶݔሻቁ ∨ ∀ݔ൫ሺܲݔ ∧ ܳݔሻ ∧ ܶݔ൯ 24) –1. ∀ݔሺݔܦ → ݔܧሻ ⊢ ∃ݔሺݔܦ → ݔܧሻ ∨ ∀ݔሺݔܧ → ݔܦሻ 25) –1. ∀ݔሺܲݔ ∨ ܳݔ → ܴݔ ∧ ܵݔሻ –2. ∃ݔሺܶݔ ∧ ܲݔሻ ⊢ ∃ݔሺܴݔ ∨ ܵݔሻ 26) –1. ∀ݔሺܲݔ ∨ ܴݔ → ܵݔ ∧ ݔܭሻ –2. ∃ݔ ሺݔܯ ∧ ܲݔሻ ⊢ ∃ݔሺܵݔ ∨ ݔܭሻ 27) –1. ∀ݔሺݔܣ → ݔܤሻ –2. ∃ݔሺݔܯ ∧ ݔܣሻ –3. ∀ݔሺݔܯ ∧ ݔܤ → ܵݔሻ ⊢ ∃ݔ൫ܵݔ ∨ ሺܳݔ ↔ ܴݔሻ൯ 28) –1. ∀ݔሺܲݔ ∧ ܳݔሻ –2. ∀ݔሺܳݔ → ܴݔሻ –3. ∀ݔሺܴݔ ∨ ܶݔ → ܵݔሻ –4. ∀ݔሺܵݔ ∨ ܶݔ → ܷݔሻ ⊢ ∀ݔ൫ܷݔ ∨ ሺܹݔ ∨ ݕܩሻ൯ 29) –1. ∀ݔܲݔ ∧ ∀ݔܳݔ –2. ∀ݔܴݔ ∧ ∀ݔܵݔ –3. ∀ݔ൫ሺܳݔ ∧ ܵݔሻ ∨ ܶݔ → ܷݔ ∧ ܴݔ൯ –4. ∀ݔሺܷݔ ∨ ܸݔ → ܹݔሻ ⊢ ∀ݔ ቀܹݔ ∧ ൫ܲݔ ∨ ሺܳݔ ↔ ܴݔሻ൯ቁ 30) –1. ∃ݔሺݔܯ ∧ ܵݔሻ –2. ∀ݔሺݔܯ ∨ ܶݔ → ܴݔሻ –3. ∀ݔሺܴݔ ∨ ሺNx ∧ ܲݔሻ → ܷݔ ∧ ܴݔሻ –4. ∀ݔሺܷݔ ∨ ܸݔ → ܳݔሻ ⊢ ∃ݔܳݔ ∨ ∀ݕݔሺܼݕ ↔ ݔܪሻ 31) –1. ∀ݔሺܲݔ ∧ ܳݔሻ ∧ ∀ݕሺܴݕ ∨ ܵݕሻ –2. ∀ݔሺܲݔ ∨ ܶݔ → ܵݔሻ –3. ∀ݔሺܴݔ ∨ ܵݔ → ܷݔሻ ⊢ ∀ݔሺܵݔ ∧ ܷݔሻ ∨ ∃ݔሺܹݔ ↔ ܼݔሻ
4.
Alfonso Cabanzo 4 32) –1. ∀ݔሺܲݔ
→ ܳݔሻ –2. ∀ݔሺܳݔ ∨ ݔܯ → ܶݔሻ –3. ∃ݕሺݕܯ ∧ ܲݕሻ ⊢ ∃ݔሺܶݔ ∧ ݔܯሻ 33) –1. ∀ݔሺݔܥ → ݔܪሻ –2. ∀ݔሺݔܪ ∨ ܶݔ → ݔܨሻ –3. ∃ ݖܥݖ ⊢ ∃ݔܨݔ ∨ ∀ݔሺݔܯ ∧ ݔܩሻ 34) –1. ∀ݔܯ(ݔ ∨ ܵݔ → ܶݔ ∧ ܳ)ݔ –2. ∀ݔሺܶݔ ∨ ܴݔ → ܵݔ ∧ ݔܬሻ –3. ∀ݔሺܵݔ ∨ ܹݔ → ݔܭ ∧ ݔܮሻ –4. ∃ݔܯݔ ∧ ∀ݔሺܲݔ → ܳݔሻ ⊢ ∃ݔܭݔ ∨ ∀ݔሺܼݔ ↔ ܹݔሻ 35) –1. ∀ݔሺܲݔ ∧ ܳݔሻ ⊢ ∀ݔሺܴݔ → ܳݔ ∨ ܵݔሻ 36) ⊢ ∀ݔሺܲݔ → ܲݔ ∨ ܳݔሻ 37) ⊢ ∀ݔሺܲݔ ∧ ܳݔ → ܲݔሻ 38) –1. ∀ݔ൫ሺݔܯ → ܵݔሻ → ܶݔ൯ –2. ∃ݔሺܵݔ ∧ ܴݔሻ ⊢ ∃ݔሺܶݔ ∧ ܴݔሻ 39) –1. ∃ݔሺܵݔ ∧ ܶݔሻ → ∀ݔܯݔ –2. ∃ݖܯݖ ∨ ∀ݔܵݔ → ∃ݔܪݔ ⊢ ∀ݔሺܵݔ ∧ ܶݔሻ → ∃ ݕܪݕ 40) –1. ∀ݔሺݔܯ ∧ ሺܶݔ ∨ ݔܪሻ → ܵݔሻ ⊢ ∀ݔ ቀݔܯ → ൫ܶݔ → ሺܵݔ ∨ ܹݔሻ൯ቁ 41) –1. ∀ݔሺݔܯ ∧ ሺܶݔ ∧ ܵݔሻ → ݔܭሻ ⊢ ∀ݔ ቀݔܯ → ൫ܶݔ → ሺܵݔ → ݔܭ ∨ ܼݔሻ൯ቁ 42) –1. ∀ݔ൫ܱݔ → ሺݔܪ → ݔܭሻ൯ → ∃ݕݔሺݕݔܬ ∧ ܵܽݔሻ –2. ∀ݔሺܱݔ ∧ ݔܪ → ݔܭሻ ⊢ ∀ݔܶݔ → ∃ݖݔݕሺݔݕܬ ∧ ܵݖݕሻ 43) ⊢ ∀ݕݔ ቀሺݕݔܣ → ݕݔܤሻ → ൫ሺݕݔܤ → ݕݔܥሻ → ሺݕݔܣ → ݕݔܥሻ൯ቁ 44) –1. ∀ݕݔ൫ݔܪ → ሺݕݔܬ → ݔݕܭሻ൯ ⊢ ∀ݕݔ൫ݔܪ ∧ ݕݔܬ → ݕݔܭ ∨ ሺܲݔ ↔ ܴݔሻ൯
5.
Alfonso Cabanzo 5 45) –1. ∀ݔሺܲݔ
→ ܵݔሻ –2. ∀ݔሺܵݔ ∨ ܶݔ → ݔܭሻ ⊢ ∀ݔሺܲݔ → ݔܭ ∨ ݔܬሻ 46) –1. ∀ݕݔ൫ሺݕݔܭ ∧ ܶݕݔሻ൯ –2. ∀ݔሺݕݔܭ → ܵݔሻ –3. ∀ݔሺܵݔ ∧ ܳݔ → ∃ݕܹݕሻ ⊢ ∀ݔሺܳݔ → ∃ݕܹݕ ∧ ܶݔሻ 47) –1. ∀ݕݔሺܲݕݔ ∧ ܳݔݕሻ –2. ∀ݔሺܳܽݔ ∧ ܵܽݔ → ܶݔሻ –3. ∀ݔ൫ሺܹݔ → ܶݔሻ → ݔܯ൯ ⊢ ∃ܽݔܵݔ → ∃ݔܯݔ 48) –1. ∀ݕݔ ቀ൫ܶݕݔ ∧ ሺܵݕݔ → ܶݕݔሻ൯ቁ –2. ∀ݔݕ ቀ൫ሺܵݔݕ → ܶݔݕሻ ∧ ݔݕܯ൯ → ݔݕܭቁ ⊢ ∀ݕݔ൫ݕݔܯ → ሺݕݔܬ → ݕݔܭሻ൯ 49) –1. ∀ݔሺܶݔ ∧ ݔܯ → ݔܬ ∨ ܼݔሻ –2. ∃ݔሺݔܯ ∧ ܵݔሻ ⊢ ∀ݔܶݔ → ∃ݔ൫ሺݔܬ ∨ ܼݔሻ ∨ ሺݔܤ ∧ ݔܥሻ൯ 50) –1. ∃ݔ൫ሺܵݔ ∧ ܲݔሻ ∧ ሺܼݔ ∧ ݕሻ൯ –2. ∀ݔ൫ሺܵݔ ∧ ܼݔሻ ∧ ሺݔܭ ∨ ݔܭሻ → ݔܣ ∨ ܹݔ൯ ⊢ ∃ݔ൫ሺݔܭ ∨ ݔܭሻ → ݔܣ ∨ ܹݔ൯
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