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First Order of Logics

In: Computers and Technology

Submitted By LinhLung
Words 1043
Pages 5
Student Name: Tran Thai Dieu Linh
Student ID: USTHBI3 - 052

I.28 (p.428) Translate the following statements into symbolic form. Avoid negation signs preceding quantifiers. The predicate letters are given in parentheses.
A good violin is rare and expensive. (G, V, R, E)
Solution:
x is a violin V(x) x is good G(x) x is rare R(x) x is expensive E(x)
→ symbolic form: ( (V(x) ^ G(x) ) ( (R(x) ^ E(x) ) I.29 (p.428) Translate the following statements into symbolic form. Avoid negation signs preceding quantifiers. The predicate letters are given in parentheses. Violins and cellos are stringed instruments. (V, C, S, I)
Solution:
x are violins V(x) x are cellos C(x) x has string S(x) x are instruments I(x)
→ symbolic form: : ( (V(x) ^ C(x) ) ( (S(x) ^ I(x) )

I.8 (p.439) Use the eighteen rules of inference to derive the conclusions of the following symbolized arguments. Do not use either conditional proof or indirect proof. 1. ∀x( A(x) B(x) )
2. A(m) ^ A(n) / B(m) ^ B(n) No. | Predicate | Note | 1. | ∀x( A(x) B(x) ) | Premise(P) | 2. | A(m) ^ A(n) | P | | B(m) ^ B(n) | Expected Conclusion | 3. | A(m) B(m) | 1, UI | 4. | A(n) B(n) | 1, UI | 5. | A(m) | 2, Com, SImp | 6. | A(n) | 2, Simp | 7. | B(m) | 3, 5, MP | 8. | B(n) | 4, 6, MP | 9. | B(m) ^ B(n) | 7, 8, Conj | | | Proved |

II.8 (p.440) Translate the following arguments into symbolic form. Then use the eighteen rules of inference to derive the conclusion of each. Do not use conditional or indirect proof Some huckleberries are ripe. Furthermore, some boysenberries are sweet. If there are any huckleberries, then the boysenberries are edible if they are sweet. Therefore, some boysenberries are edible. (H, R, B, S, E)
Solution:
H(x) x are hukleberries R(x) x is ripe B(x) x are boysenberries S(x) x is sweet E(x) x is edible NO. | English | Predicates | 1. | Some huckleberries are ripe | ∃x (H(x) ^ R(x)) | 2. | Some boysenberries are sweet | ∃x (B(x) ^ S(x)) | 3. | The boysenberries are edible if they are sweet | (B(x)^S(x) ) E(x) | 4. | If there are any huckleberries, then the boysenberries are edible if they are sweet. | ∃xH(x) ∀x [[B(x) ^S(x)] E(x) ] | 5. | Some boysenberries are edible | ∃x (B(x) ^ E(x)) |
Prove:
No. | Predicate | Note | 1. | ∃x (H(x) ^ R(x)) | Premise(P) | 2. | ∃x (B(x) ^ S(x)) | P | 3. | ∃xH(x) ∀x [[B(x) ^S(x)] E(x) ] | P | | ∃x (B(x) ^ E(x) ) | Expected Conclusion | 4. | H(m) ^ R(m) | 1, EI | 5. | B(n) ^ S(n) | 2, EI | 6. | H(m) | 4, Com, Simp | 7. | ∃x H(x) | 6, EG | 8. | ∀x (B(x) ^ S(x) ) E(x) | 3, 7, MP | 9. | ( B(n) ^ S(n) ) E(n) | 8, UI | 10. | E(n) | 5, 9, MP | 11. | B(n) | 5, Com, Simp | 12. | B(n) ^ E(n) | 10, 11, Conj | 13. | ∃x (B(x) ^ E(x) ) | 12, EG | | | Proved |

I.8 (p.443) Use the change of quantifier rules together with the eighteen rules of inference to derive the conclusions of the following symbolized arguments. Do not use either conditional proof or indirect proof.
1. ∀x A(x) ∃x ¬B(x)
2. ¬ ∀xB(x) ∃x¬ C(x) / ∀x C(x) ∃x¬ A(x)
Solution:
No. | Predicate | Note | 1. | ∀x A(x) ∃x ¬B(x) | Premise(P) | 2. | ¬ ∀xB(x) ∃x¬ C(x) | P | | ∀x C(x) ∃x¬ A(x) | Expected Conclusion | 3. | ¬ ∀xA(x) ¬ ∀x B(x) | 1, CQ | 4. | ∀x A(x) ∃x¬ C(x) | 2, 3, HS | 5. | ¬ ∀xA(x) v ∃x¬ C(x) | 4, Impl | 6. | ¬ ∀x C(x) v ∃x ¬A(x) | 5, CQ, Com | 7. | ∀x C(x) ∃x¬ A(x) | 6, Impl | | | Proved |

II.2 (p.444) Translate the following arguments into symbolic form. Then use the change of quantifier rules and the eighteen rules of inference to derive the conclusion of each. Do not use either conditional proof or indirect proof. Either Dr. Adams is an internist or all the pathologists are internists. But it is not the case that there are any internists. Therefore, Dr. Adams is not a pathologist. (I, P)
Solution:
No. | English | Predicates | 1. | x is an internist | I(x) | 2. | x is a pathologist | P(x) | 3. | All the pathologists are internists | ∀x [P(x) I(x) ] | 4. | There are any internists | ∃x I(x) | | So, the following statements … | …mean: | 5. | Either Dr. Adams is an internist or all the pathologists are internists | I(Dr.Adams) v ∀x [P(x) I(x) ] | 6. | it is not the case that there are any internists | ¬∃x I(x) | 7. | Dr. Adams is not a pathologist | ¬P(Dr.Adams) |

Proved: No. | Predicate | Note | 1. | I(Dr.Adams) v ∀x [P(x) I(x) ] | Premise(P) | 2. | ¬∃x I(x) | P | | ¬P(Dr.Adams) | Expected Conclusion | 3. | ∀x ¬I(x) | 2, CQ | 4. | ¬I(Dr. Adams) | 3, UI | 5. | ∀x [P(x) I(x) ] | 1, 4, Com, DS | 6. | P(Dr.Adams) I (Dr. Adams) | 5, UI | 7. | ¬P(Dr.Adams) v I (Dr. Adams) | 6, Impl | 8. | ¬P(Dr.Adams) | 4, 7, DS | | | Proved |

I.13 (p.449) Use either indirect proof or conditional proof to derive the conclusions of the following symbolized arguments: 1. ∃xA(x) ∀ x(B(x) C(x) )
2. ∃xD(x) ∃xB(x) / ∃x(A(x) ^ D(x)) ∃xC(x) No. | Predicate | Note | 1. | ∃xA(x) ∀ x(B(x) C(x) ) | Premise(P) | 2. | ∃xD(x) ∃xB(x) | P | | ∃x(A(x) ^ D(x)) ∃xC(x) | Expected Conclusion | | 3. ∃x(A(x) ^ D(x)) | ACP | | 4. A(m) ^ D(m) | 3, EI | | 5. A(m) | 4, Com, Simp | | 6. ∃xA(x) | 5, EG | | 7. ∀ x(B(x) C(x) ) | 1, 6, MP | | 8. D(m) | 4, Simp | | 9. ∃xD(x) | 8, EG | | 10. ∃xB(x) | 2, 9, MP | | 11. B(n) | 10, EI | | 12. B(n) C(n) | 7, UI | | 13. C(n) | 11, 12, MP | | 14. ∃xC(x) | 13, EG | 15. | ∃x [(A(x) ^ D(x)] ∃xC(x) | 3-14, CP |…...

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