湖北大学2004-2005学年度第二学期课程考试试题纸(第1页共4页) 长黃兴 湖北课程考试试题纸 兴兴兴 课程名称: 数理逻辑 (B卷) 考试方式: (闭卷) 印刷份数:30 学院 哲学系任课教师:_宋伟 专业年级 哲学2003级 兴兴兴兴兴兴兴 三|三|四五六七|八阅卷 教师 得分…得分 Complete the following blanks. (each 2 points, total 20 points) 製 *1. For a statement form involving n different statement variables, the truth table will have *2. if A and B are statement forms, A B if(A-B)is a tautology. *3. For any statement forms A and B, ((AAB)is logically equivalent to *4. The Soundness Theorem for the formal system L: 5. Translate the statement 'NO number is both odd and even. into symbols *6.Any formal system consists of a 7. The formal system L is i.e. there is an effective method for deciding, given any Ak*wfof L, whether it is a theorem ofL 排鄭*|8 f an occurrence of a variable in a wf. is not bound it is said to be *9. A set is if it can be put in one-one correspondence with the set of natural ***numbers 0(3x1)((V X2)A(x1, X2)-(3x3)A,(x1, x3))yields a Skolemised form 兴兴 Solve to following problems. (each 10 points, total 60 points) 分 1. Find the disjunctive normal form and the conjunctive normal form which are logically quivalent to(((p)Vqr) 说明:本试卷将作为样卷直接制版胶印,请命题教师在试题之间留足答题空间
湖北大学 2004—2005 学年度第二学期课程考试 试题纸（第 1 页 共 4 页） 说明： 本试卷将作为样卷直接制版胶印，请命题教师在试题之间留足答题空间。 课程考试试题纸 课程名称： 数理逻辑 （ B 卷） 考试方式： （闭卷） 印刷份数： 30 学 院： 哲 学 系 任课教师： 宋 伟 专业年级： 哲学 2003 级 题 号 一 二 三 四 五 六 七 八 总分 阅卷 教师 得 分 …………………………………………………………………………………………………… 一. Complete the following blanks.(each 2 points, total 20 points) 1. For a statement form involving n different statement variables, the truth table will have rows 2. if A and B are statement forms, A B if (A→B) is a tautology. 3. For any statement forms A and B, (～(A∧B)) is logically equivalent to . 4. The Soundness Theorem for the formal system L: . 5. Translate the statement ‘NO number is both odd and even.’ into symbols: . 6. Any formal system consists of a and a . 7. The formal system L is , i.e. there is an effective method for deciding, given any wf. of L, whether it is a theorem of L. 8. If an occurrence of a variable in a wf. is not bound it is said to be . 9. A set is if it can be put in one-one correspondence with the set of natural numbers. 10. ( x1) (( x2)A 2 1 (x1,x2) →( x3)A 2 2 (x1,x3)) yields a Skolemised form: . 二. Solve to following problems. (each 10 points, total 60 points) 1. Find the disjunctive normal form and the conjunctive normal form which are logically equivalent to (((～p)∨q) →r). 得 分 得 分 * * * * * * * * * * * 学 号： 姓 名： * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 学 院： 专业年级： * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
湖北大学2004-2005学年度第二学期课程考试试题纸(第2页共4页) 2. Find a statement form involving only I which is logically equivalent to(p-g) 3. Determine whether the argument form((p)-(-q), q;.. p is valid or invalid 4. Using the Deduction Theorem for L, show that the following wf is theorem of l, where A and b are any wfs of L (A→(B→C)→(B→(A→C)) 说明:本试卷将作为样卷直接制版胶印,请命题教师在试題之间留足答题空间
湖北大学 2004—2005 学年度第二学期课程考试 试题纸（第 2 页 共 4 页） 说明： 本试卷将作为样卷直接制版胶印，请命题教师在试题之间留足答题空间。 2. Find a statement form involving only | which is logically equivalent to (p→q). 3.Determine whether the argument form ((~p)→ (~q)), q; ∴p is valid or invalid. 4. Using the Deduction Theorem for L, show that the following wf. is theorem of L, where A and B are any wfs. of L. ((A→(B→C)) →(B→(A→C)))
湖北大学2004-2005学年度第二学期课程考试试题纸(第3页共4页) 5. Find an interpretation in which the wf(V x1)(A(x)-AH(f(xD) is interpreted by a statement which is false =.Proof.(each 10 points, total 30 points 1. Prove that if the statement form(A1∧…∧An)→A) is a tautology then the argument form ∴ A is valid 说明:本试卷将作为样卷直接制版胶印,请命题教师在试題之间留足答题空间
湖北大学 2004—2005 学年度第二学期课程考试 试题纸（第 3 页 共 4 页） 说明： 本试卷将作为样卷直接制版胶印，请命题教师在试题之间留足答题空间。 5. Find an interpretation in which the wf. ( x1) (A 1 1 (x1) →A 1 1 (f 1 1 (x1))) is interpreted by a statement which is false. 三. Proof. (each 10 points, total 30 points) 1. Prove that if the statement form ((A1∧…∧An) → A) is a tautology then the argument form A1, A2, …, An ; ∴A is valid. 得 分
湖北大学2004-2005学年度第二学期课程考试试题纸(第4页共4页) 2. Prove that is B is a contradiction then B cannot be a theorem of any consistent extension of L 3. Prove that if, in a particular interpretation I, the wfs. A and(a-B)are true then b is also 说明:本试卷将作为样卷直接制版胶印,请命题教师在试題之间留足答题空间
湖北大学 2004—2005 学年度第二学期课程考试 试题纸（第 4 页 共 4 页） 说明： 本试卷将作为样卷直接制版胶印，请命题教师在试题之间留足答题空间。 2. Prove that is B is a contradiction then B cannot be a theorem of any consistent extension of L. 3. Prove that if, in a particular interpretation I, the wfs. A and (A→B) are true, then B is also true