【头歌educoder】离散数学实训参考-第四章-数理逻辑-part2-自然推理系统

目录

第一关

第二关

第三关

第四关

第五关

第六关

第七关

第八关

第九关

第十关 

第十一关

第十二关

第十三关

第十四关

第十五关

第十六关

第十七关

第十八关

第十九关

第二十关

第二十一关

第二十二关

第二十三关

第二十四关

第二十五关

第二十六关

第二十七关

第二十八关

第二十九关

第三十关

第三十一关

第三十二关

第三十三关

第三十四关

第三十五关

第三十六关

第三十七关

第三十八关 


         本着来熟悉python的,遇到这节证明题目,于是尝试以下。较难的题是23题和29题,这两题都使用了导出规则;以及20题会显得吓人。


第一关

左推右、右推左、则左<->右;L=\overline{A}\, \overline{B}C;R=\overline{B}\, \overline{A}C

Premise:

Conclusion: (A->(B->C))<->(B->(A->C))

1.  A->(B->C),B,A   |-  A   prem

2.  A->(B->C),B,A   |-  B   prem

3.  A->(B->C),B,A   |-  A->(B->C)   prem

4.  A->(B->C),B,A   |-  B->C    imple 1,3

5.  A->(B->C),B,A   |-  C   imple 2,4

6.  A->(B->C),B |-  A->C    impli 5

7.  A->(B->C)   |-  B->(A->C)   impli 6

8.  B->(A->C),A,B   |-  A   prem

9.  B->(A->C),A,B   |-  B   prem

10. B->(A->C),A,B   |-  B->(A->C)   prem

11. B->(A->C),A,B   |-  A->C    imple 9,10

12. B->(A->C),A,B   |-  C   imple 8,11

13. B->(A->C),A |-  B->C    impli 12

14. B->(A->C)   |-  A->(B->C)   impli 13

15. |-  (A->(B->C))<->(B->(A->C))   equivi 7,14

第二关

并的交换律;A\cup B \cup C = A\cup (B\cup C)

Premise: (A\/B)\/C

Conclusion: A\/(B\/C)

1.  A   |-  A   prem

2.  A   |-  A\/(B\/C)   ori 1

3.  B   |-  B   prem

4.  B   |-  B\/C    ori 3

5.  B   |-  A\/(B\/C)   ori 4

6.  C   |-  C   prem

7.  C   |-  B\/C    ori 6

8.  C   |-  A\/(B\/C)   ori 7

9.  A\/B    |-  A\/(B\/C)   ore 2,5

10. (A\/B)\/C   |-  A\/(B\/C)   ore 9,8 

第三关

并的交换律;A \cup (B\cup C) = A\cup B \cup C。 

Premise: A\/(B\/C)

Conclusion: (A\/B)\/C

1.  C   |-  C   prem

2.  C   |-  (A\/B)\/C   ori 1

3.  B   |-  B   prem

4.  B   |-  A\/B    ori 3

5.  B   |-  (A\/B)\/C   ori 4

6.  A   |-  A   prem

7.  A   |-  A\/B    ori 6

8.  A   |-  (A\/B)\/C   ori 7

9.  B\/C    |-  (A\/B)\/C   ore 2,5

10. A\/(B\/C)   |-  (A\/B)\/C   ore 9,8

第四关

命题取反,蕴含指向也取反;L=\overline{A}\,B;R=\overline{\overline{B}}\,\overline{A}

Premise:

Conclusion: (A->B)<->(~B -> ~A)

1.  A->B,~B,A   |-  ~B  prem

2.  A->B,~B,A   |-  A   prem

3.  A->B,~B,A   |-  A->B    prem

4.  A->B,~B,A   |-  B   imple 2,3

5.  A->B,~B |-  ~A  ni 1,4

6.  A->B    |-  ~B -> ~A    impli 5

7.  ~B -> ~A,A,~B   |-  A   prem

8.  ~B -> ~A,A,~B   |-  ~B  prem

9.  ~B -> ~A,A,~B   |-  ~B -> ~A    prem

10  ~B -> ~A,A,~B   |-  ~A  imple 8,9

11  ~B -> ~A,A  |-  ~(~B)   ni 7,10

12. ~B -> ~A,A  |-  B   nne 11

13. ~B -> ~A    |-  A->B    impli 12

14. |-  (A->B)<->(~B -> ~A) equivi 6,13 

第五关

与自身取反求交;A\cap\overline{A} 永假。

Premise: A/\~A

Conclusion: F

1.  A/\~A   |-  A/\~A   prem

2.  A/\~A   |-  A   ande 1

3.  A/\~A   |-  ~A  ande 1

4.  A/\~A   |-  F   ne 2,3 

第六关

与自身取反求交;A\cap\overline{A} 永假。 

Premise: F

Conclusion: A/\~A

1.  F   |-  F   prem

2.  F   |-  ~F  fi

3.  F   |-  A/\~A   ne 1,2 

第七关

Pe=\overline{A}\,\overline{B}C;Co=\overline{AB}C=(A\overline{B}+\overline{A}B+\overline{A}\,\overline{B})C

Premise: A->(B->C)

Conclusion: A/\B->C

1.  A->(B->C),A/\B  |-  A/\B    prem

2.  A->(B->C),A/\B  |-  A   ande 1

3.  A->(B->C),A/\B  |-  B   ande 1

4.  A->(B->C),A/\B  |-  A->(B->C)   prem

5.  A->(B->C),A/\B  |-  B->C    imple 2,4

6.  A->(B->C),A/\B  |-  C   imple 3,5

7.  A->(B->C)   |-  A/\B->C impli 6

第八关

Pr = \overline{AB}C;Co = \overline{A}\,\overline{B}C

Premise: A/\B->C

Conclusion: A->(B->C)

1.  A/\B->C,A,B |-  A   prem

2.  A/\B->C,A,B |-  B   prem

3.  A/\B->C,A,B |-  A/\B->C prem

4.  A/\B->C,A,B |-  A/\B    andi 1,2

5.  A/\B->C,A,B |-  C   imple 3,4

6.  A/\B->C,A   |-  B->C    impli 5

7.  A/\B->C |-  A->(B->C)   impli 6

第九关

量词,这些题目都是显然的,主要是如何用给定的描述系统写过程

Premise:(All x)P(x)

Conclusion:(Exist x)P(x)

1.  (All x)P(x) |-  (All x)P(x) prem

2.  (All x)P(x) |-  P(a)    ae 1 (a/x)

3.  (All x)P(x) |-  (Exist x)P(x)   ei 2 (a/x)

第十关 

Premise: (Exist x)(All y)P(x,y)

Conclusion: (All y)(Exist x)P(x,y)

1.  (All y)P(x,y)   |-  (All y)P(x,y)   prem

2.  (All y)P(x,y)   |-  P(x,b)  ae 1 (b/y)

3.  (All y)P(x,y)   |-  (Exist x)P(x,b) ei 2 (x/x)

4.  (All y)P(x,y)   |-  (All y)(Exist x)P(x,y)  ai 3

5.  (Exist x)(All y)P(x,y)  |-  (All y)(Exist x)P(x,y)  lei 4

第十一关

Premise: (All x)P(x)\/(All x)Q(x)

Conclusion: (All x)(P(x)\/Q(x))

1.  (All x)P(x) |-  (All x)P(x) prem

2.  (All x)P(x) |-  P(b)    ae 1 (b/x)

3.  (All x)P(x) |-  P(b)\/Q(b)  ori 2

4.  (All x)Q(x) |-  (All x)Q(x) prem

5.  (All x)Q(x) |-  Q(b)    ae 4 (b/x)

6.  (All x)Q(x) |-  P(b)\/Q(b)  ori 5

7.  (All x)P(x)\/(All x)Q(x)    |-  P(b)\/Q(b)  ore 3,6

8.  (All x)P(x)\/(All x)Q(x)    |-   (All x)(P(x)\/Q(x))    ai 7

第十二关

Premise: (All x)(A->P(x))

Conclusion: A->(All x)P(x)

1.  (All x)(A->P(x)),A  |-  (All x)(A->P(x))    prem

2.  (All x)(A->P(x)),A  |-  A->P(b) ae 1 (b/x)

3.  (All x)(A->P(x)),A  |-  A   prem

4.  (All x)(A->P(x)),A  |-  P(b)    imple 3,2

5.  (All x)(A->P(x)),A  |-  (All x)P(x) ai 4

6.  (All x)(A->P(x))    |-  A->(All x)P(x)  impli 5

第十三关

Premise: A->(All x)P(x)

Conclusion: (All x)(A->P(x))

1.  A->(All x)P(x),A    |-  A   prem

2.  A->(All x)P(x),A    |-  A->(All x)P(x)  prem

3.  A->(All x)P(x),A    |-  (All x)P(x) imple 1,2

4.  A->(All x)P(x),A    |-  P(x)    ae 3 (x/x)

5.  A->(All x)P(x)  |-  A->P(x) impli 4

6.  A->(All x)P(x)  |-  (All x)(A->P(x))    ai 5

第十四关

Premise: (Exist x)(P(x)\/Q(x))

Conclusion: (Exist x)P(x)\/(Exist x)Q(x)

1.  P(x)    |-  P(x)    prem

2.  Q(x)    |-  Q(x)    prem

3.  P(x)    |-  (Exist x)P(x)   ei 1 (x/x)

4.  Q(x)    |-  (Exist x)Q(x)   ei 2 (x/x)

5.  P(x)    |-  (Exist x)P(x)\/(Exist x)Q(x)    ori 3

6.  Q(x)    |-  (Exist x)P(x)\/(Exist x)Q(x)    ori 4

7.  P(x)\/Q(x)  |-  (Exist x)P(x)\/(Exist x)Q(x)    ore 5,6

8.  (Exist x)(P(x)\/Q(x))   |-  (Exist x)P(x)\/(Exist x)Q(x)    lei 7

第十五关

Premise: (Exist x)P(x)\/(Exist x)Q(x)

Conclusion: (Exist x)(P(x)\/Q(x))

1.  P(x)    |-  P(x)    prem

2.  Q(x)    |-  Q(x)    prem

3.  P(x)    |-  P(x)\/Q(x)  ori 1

4.  Q(x)    |-  P(x)\/Q(x)  ori 2

5.  (Exist x)P(x)   |-  (Exist x)(P(x)\/Q(x))   eei 3

6.  (Exist x)Q(x)   |-  (Exist x)(P(x)\/Q(x))   eei 4

7.  (Exist x)P(x)\/(Exist x)Q(x)    |-  (Exist x)(P(x)\/Q(x))   ore 5,6

第十六关

Premise: (All x)(P(x)/\Q(x))

Conclusion: (All x)P(x)/\(All x)Q(x)

1.  P(x)/\Q(x)  |-  P(x)/\Q(x)  prem

2.  P(x)/\Q(x)  |-  P(x)    ande 1

3.  P(x)/\Q(x)  |-  Q(x)    ande 1

4.  (All x)(P(x)/\Q(x)) |-  (All x)P(x) aai 2

5.  (All x)(P(x)/\Q(x)) |-  (All x)Q(x) aai 3

6.  (All x)(P(x)/\Q(x)) |-  (All x)P(x)/\(All x)Q(x)    andi 4,5

第十七关

Premise: (All x)P(x)/\(All x)Q(x)

Conclusion: (All x)(P(x)/\Q(x))

1.  (All x)P(x)/\(All x)Q(x)    |-  (All x)P(x)/\(All x)Q(x)    prem

2.  (All x)P(x)/\(All x)Q(x)    |-  (All x)P(x) ande 1

3.  (All x)P(x)/\(All x)Q(x)    |-  (All x)Q(x) ande 1

4.  (All x)P(x)/\(All x)Q(x)    |-  P(t)    ae 2 (t/x)

5.  (All x)P(x)/\(All x)Q(x)    |-  Q(t)    ae 3 (t/x)

6.  (All x)P(x)/\(All x)Q(x)    |-  P(t)/\Q(t)  andi 4,5

7.  (All x)P(x)/\(All x)Q(x)    |-  (All x)(P(x)/\Q(x)) ai 6

第十八关

Premise: (Exist x)P(x)->(All x)Q(x)

Conclusion: (All x)(P(x)->Q(x))

1.  (Exist x)P(x)->(All x)Q(x),P(x) |-  (Exist x)P(x)->(All x)Q(x)  prem

2.  (Exist x)P(x)->(All x)Q(x),P(x) |-  P(x)    prem

3.  (Exist x)P(x)->(All x)Q(x),P(x) |-  (Exist x)P(x)   ei 2 (x/x)

4.  (Exist x)P(x)->(All x)Q(x),P(x) |-  (All x)Q(x) imple 3,1

5.  (Exist x)P(x)->(All x)Q(x),P(x) |-  Q(x)    ae 4 (x/x)

6.  (Exist x)P(x)->(All x)Q(x)  |-  P(x)->Q(x)  impli 5

7.  (Exist x)P(x)->(All x)Q(x)  |-  (All x)(P(x)->Q(x)) ai 6

第十九关

Premise: (All x)(P(x)->Q(x))

Conclusion: (All x)P(x)->(All x)Q(x)

1.  (All x)(P(x)->Q(x)),(All x)P(x) |-  (All x)P(x) prem

2.  (All x)(P(x)->Q(x)),(All x)P(x) |-  (All x)(P(x)->Q(x)) prem

3.  (All x)(P(x)->Q(x)),(All x)P(x) |-  P(x)    ae 1 (x/x)

4.  (All x)(P(x)->Q(x)),(All x)P(x) |-  P(x)->Q(x)  ae 2 (x/x)

5.  (All x)(P(x)->Q(x)),(All x)P(x) |-  Q(x)    imple 3,4

6.  (All x)(P(x)->Q(x)),(All x)P(x) |-  (All x)Q(x) ai 5

7.  (All x)(P(x)->Q(x)) |-  (All x)P(x)->(All x)Q(x)    impli 6

第二十关

这题目有点吓人,虚张声势;注意ee那条规则运用时,不要出现代替量在前提之类的错误。

Premise: (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y)

Conclusion: (All x)R(x,x)

1.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y) |-  (All x)(Exist y)R(x,y)  prem

2.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y) |-  (Exist y)R(x,y) ae 1 (x/x)

3.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  R(x,b)  prem

4.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  (All x)(All y)(R(x,y)->R(y,x))  prem

5.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  (All y)(R(x,y)->R(y,x)) ae 4 (x/x)

6.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  R(x,b)->R(b,x)  ae 5 (b/y)

7.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  R(b,x)  imple 3,6

8.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  R(x,b)/\R(b,x)  andi 3,7

9.  (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  (All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z))   prem

10. (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  (All y)(All z)(R(x,y)/\R(y,z)->R(x,z))  ae 9 (x/x)

11. (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  (All z)(R(x,b)/\R(b,z)->R(x,z)) ae 10 (b/y)

12. (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  R(x,b)/\R(b,x)->R(x,x)  ae 11 (x/z)

13. (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y),R(x,b)  |-  R(x,x)  imple 8,12

14. (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y) |-  R(x,x)  ee 2,13

15. (All x)(All y)(R(x,y)->R(y,x)),(All x)(All y)(All z)(R(x,y)/\R(y,z)->R(x,z)),(All x)(Exist y)R(x,y) |-  (All x)R(x,x)   ai 14

第二十一关

应该是叫摩根定律;\overline{AB}=\overline{A}+\overline{B}

Premise:

Conclusion: ~(A/\B)<->(~A\/~B)

1.  ~A,A/\B |-  A/\B    prem

2.  ~A,A/\B |-  A   ande 1

3.  ~A,A/\B |-  ~A  prem

4.  ~A  |-  ~(A/\B) ni 2,3

5.  ~B,A/\B |-  A/\B    prem

6.  ~B,A/\B |-  B   ande 5

7.  ~B,A/\B |-  ~B  prem

8.  ~B  |-  ~(A/\B) ni 6,7

9.  ~A\/~B  |-  ~(A/\B) ore 4,8

10. ~(A/\B),A,B |-  ~(A/\B) prem

11  ~(A/\B),A,B |-  A   prem

12. ~(A/\B),A,B |-  B   prem

13. ~(A/\B),A,B |-  A/\B    andi 11,12

14. ~(A/\B),A   |-  ~B  ni 10,13

15. ~(A/\B),A   |-  ~A\/~B  ori 14

16. ~(A/\B),~A  |-  ~A  prem

17. ~(A/\B),~A  |-  ~A\/~B  ori 16

18. ~(A/\B) |-  ~A\/~B  preme 15,17

19. |-  ~(A/\B)<->(~A\/~B)  equivi 9,18

第二十二关

分配律。

Premise:

Conclusion: A/\(B\/C)<->(A/\B)\/(A/\C)

1.  A/\B    |-  A/\B    prem

2.  A/\B    |-  A   ande 1

3.  A/\B    |-  B   ande 1

4.  A/\B    |-  B\/C    ori 3

5.  A/\B    |-  A/\(B\/C)   andi 2,4

6.  A/\C    |-  A/\C    prem

7.  A/\C    |-  A   ande 6

8.  A/\C    |-  C   ande 6

9.  A/\C    |-  B\/C    ori 8

10. A/\C    |-  A/\(B\/C)   andi 7,9

11. (A/\B)\/(A/\C)  |-  A/\(B\/C)   ore 5,10

12. A/\(B\/C)   |-  A/\(B\/C)   prem

13. A/\(B\/C)   |-  A   ande 12

14. A/\(B\/C)   |-  B\/C    ande 12

15. A/\(B\/C),B |-  B   prem

16. A/\(B\/C),B |-  A   premi 13

17. A/\(B\/C),B |-  A/\B    andi 15,16

18. A/\(B\/C),B |-  (A/\B)\/(A/\C)  ori 17

19. A/\(B\/C),C |-  C   prem

20. A/\(B\/C),C |-  A   premi 13

21. A/\(B\/C),C |-  A/\C    andi 19,20

22. A/\(B\/C),C |-  (A/\B)\/(A/\C)  ori 21

23. A/\(B\/C),B\/C  |-  (A/\B)\/(A/\C)  ore 18,22

24. A/\(B\/C)   |-  (B\/C)->(A/\B)\/(A/\C)  impli 23

25. A/\(B\/C)   |-  (A/\B)\/(A/\C)  imple 14,24

26. |-  A/\(B\/C)<->(A/\B)\/(A/\C)  equivi 11,25

第二十三关

分配律2;左边到右边简单,右边到左边困难,使用了给的导出规则

Premise:

Conclusion: A\/(B/\C)<->(A\/B)/\(A\/C)

1.  A   |-  A   prem

2.  A   |-  A\/B    ori 1

3.  A   |-  A\/C    ori 1

4.  A   |-  (A\/B)/\(A\/C)  andi 2,3

5.  B/\C    |-  B/\C    prem

6.  B/\C    |-  B   ande 5

7.  B/\C    |-  C   ande 5

8.  B/\C    |-  A\/B    ori 6

9.  B/\C    |-  A\/C    ori 7

10. B/\C    |-  (A\/B)/\(A\/C)  andi 8,9

11. A\/(B/\C)   |-  (A\/B)/\(A\/C)  ore 4,10

12. (A\/B)/\(A\/C),~A,~B    |-  ~(A\/B) dr

13. (A\/B)/\(A\/C),~A,~B    |-  (A\/B)/\(A\/C)  prem

14. (A\/B)/\(A\/C),~A,~B    |-  A\/B    ande 13

15. (A\/B)/\(A\/C),~A   |-  ~~B ni 12,14

16. (A\/B)/\(A\/C),~A   |-  B   nne 15

17. (A\/B)/\(A\/C),~A,~C    |-  ~(A\/C) dr

18. (A\/B)/\(A\/C),~A,~C    |-  (A\/B)/\(A\/C)  prem

19. (A\/B)/\(A\/C),~A,~C    |-  A\/C    ande 18

20. (A\/B)/\(A\/C),~A   |-  ~~C ni 17,19

21. (A\/B)/\(A\/C),~A   |-  C   nne 20

22. (A\/B)/\(A\/C),~A   |-  B/\C    andi 16,21

23. (A\/B)/\(A\/C),~A   |-  A\/(B/\C)   ori 22

24. (A\/B)/\(A\/C),A    |-  A   prem

25. (A\/B)/\(A\/C),A    |-  A\/(B/\C)   ori 24

26. (A\/B)/\(A\/C)  |-  A\/(B/\C)   preme 25,23

27. |-  A\/(B/\C)<->(A\/B)/\(A\/C)  equivi 11,26 

第二十四关

与非的并。

Premise:

Conclusion: A\/F<->A

1.  A   |-  A   prem

2.  F   |-  F   prem

3.  F   |-  ~F  fi

4.  F   |-  A   ne 2,3

5.  A\/F    |-  A   ore 1,4

6.  A   |-  A\/F    ori 1

7.  |-  A\/F<->A    equivi 5,6 

第二十五关

与是的交。

Premise:

Conclusion: A/\T<->A

1.  A/\T    |-  A/\T    prem

2.  A/\T    |-  A   ande 1

3.  A   |-  A   prem

4.  A   |-  T   ti

5.  A   |-  A/\T    andi 3,4

6.  |-  A/\T<->A    equivi 2,5

第二十六关

L=\overline{A}B;R=\overline{A}+B=\overline{A} (B+\overline{B})+(\overline{A}+A)B

Premise:

Conclusion: (A->B)<->~A\/B

1.  B,A |-  B   prem

2.  B   |-  A->B    impli 1

3.  ~A,A    |-  A   prem

4.  ~A,A    |-  ~A  prem

5.  ~A,A    |-  B   ne 3,4

6.  ~A  |-  A->B    impli 5

7.  ~A\/B   |-  A->B    ore 2,6

8.  A->B,A  |-  A   prem

9.  A->B,A  |-  A->B    prem

10. A->B,A  |-  B   imple 8,9

11. A->B,A  |-  ~A\/B   ori 10

12. A->B,~A |-  ~A  prem

13. A->B,~A |-  ~A\/B   ori 12

14. A->B    |-  ~A\/B   preme 11,13

15. |-  (A->B)<->~A\/B  equivi 7,14 

第二十七关

Premise: ~(A->B)

Conclusion: A

1.  ~(A->B),~A,A    |-  ~A  prem

2.  ~(A->B),~A,A    |-  A   prem

3.  ~(A->B),~A,A    |-  A->B    ne 1,2

4.  ~(A->B),~A,A    |-  B   imple 2,3

5.  ~(A->B),~A  |-  A->B    impli 4

6.  ~(A->B),~A  |-  ~(A->B) prem

7.  ~(A->B) |-  ~(~A)   ni 5,6

8.  ~(A->B) |-  A   nne 7

第二十八关

Premise: ~(A->B)

Conclusion: ~B

1.  ~(A->B),B,A |-  B   prem

2.  ~(A->B),B   |-  A->B    impli 1

3.  ~(A->B),B   |-  ~(A->B) prem

4.  ~(A->B) |-  ~B  ni 2,3 

第二十九关

这题的右到左也是用到导出规则,学到的新东西就是把结论的非拿来当前提,然后推出矛盾,也就是反证法

Premise:

Conclusion: ((Exist x)(C->P(x)))<->(C->(Exist x)P(x))

1.  C->P(x),C   |-  C->P(x) prem

2.  C->P(x),C   |-  C   prem

3.  C->P(x),C   |-  P(x)    imple 2,1

4.  C->P(x),C   |-  (Exist x)P(x)   ei 3 (x/x)

5.  C->P(x) |-  C->(Exist x)P(x)    impli 4

6.  (Exist x)(C->P(x))  |-  C->(Exist x)P(x)    lei 5

7.  C->(Exist x)P(x),~(Exist x)(C->P(x)),C  |-  C   prem

8.  C->(Exist x)P(x),~(Exist x)(C->P(x)),C  |-  C->(Exist x)P(x)    prem

9.  C->(Exist x)P(x),~(Exist x)(C->P(x)),C  |-  (Exist x)P(x)   imple 7,8

10. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  ~(Exist x)(C->P(x)) prem

11. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  (All x)(~(C->P(x))) dr

12. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  ~(C->P(x))  ae 11 (x/x)

13. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  C/\~P(x)    dr

14. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  C   ande 13

15. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  ~P(x)   ande 13

16. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  (All x)(~P(x))  ai 15

17. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  ~(Exist x)P(x)  dr

18. C->(Exist x)P(x),~(Exist x)(C->P(x)),C  |-  ~(Exist x)P(x)  premi 17

19. C->(Exist x)P(x),~(Exist x)(C->P(x))    |-  ~C  ni 9,18

20. C->(Exist x)P(x)    |-  ~(~(Exist x)(C->P(x)))  ni 14,19

21. C->(Exist x)P(x)    |-  (Exist x)(C->P(x))  nne 20

21. |-  ((Exist x)(C->P(x)))<->(C->(Exist x)P(x))   equivi 6,21 

第三十关

Premise: (Exist x)(P(x)->C)

Conclusion: (All x)P(x)->C

1.  P(x)->C,P(x)    |-  P(x)    prem

2.  P(x)->C,P(x)    |-  P(x)->C prem

3.  P(x)->C,P(x)    |-  C   imple 1,2

4.  P(x)->C,(All x)P(x) |-  C   lai 3

5.  P(x)->C |-  (All x)P(x)->C  impli 4

6.  (Exist x)(P(x)->C)  |-  (All x)P(x)->C  lei 5 

第三十一关

Premise: (All x)P(x)->C

Conclusion: (Exist x)(P(x)->C)

1.  (All x)P(x)->C,P(b) |   (All x)P(x)->C  prem

2.  (All x)P(x)->C,P(b) |   P(b)    prem

3.  (All x)P(x)->C,P(b) |   (All x)P(x) ai 2

4.  (All x)P(x)->C,P(b) |   C   imple 1,3

5.  (All x)P(x)->C  |   P(b)->C impli 4

6.  (All x)P(x)->C  |   (Exist x)(P(x)->C)  ei 5 (b/x) 

第三十二关

Premise: (Exist x)(P(x)\/C)

Conclusion: (Exist x)P(x)\/C

1.  P(x)    |-  P(x)    prem

2.  C   |-  C   prem

3.  P(x)    |-  (Exist x)P(x)   ei 1 (x/x)

4.  P(x)    |-  (Exist x)P(x)\/C    ori 3

5.  C   |-  (Exist x)P(x)\/C    ori 2

6.  P(x)\/C |-  (Exist x)P(x)\/C    ore 4,5

7.  (Exist x)(P(x)\/C)  |-  (Exist x)P(x)\/C    lei 6

第三十三关

Premise: (Exist x)P(x)\/C

Conclusion: (Exist x)(P(x)\/C)

1.  P(x)    |-  P(x)    prem

2.  C   |-  C   prem

3.  C   |-  P(x)\/C ori 2

4.  C   |-  (Exist x)(P(x)\/C)  ei 3 (x/x)

5.  P(x)    |-  P(x)\/C ori 1

6.  (Exist x)P(x)   |-  (Exist x)(P(x)\/C)  eei 5

7.  (Exist x)P(x)\/C    |-  (Exist x)(P(x)\/C)  ore 4,6

第三十四关

Premise: (All x)(P(x)/\C)

Conclusion: (All x)P(x)/\C

1.  P(x)/\C |-  P(x)/\C prem

2.  P(x)/\C |-  P(x)    ande 1

3.  P(x)/\C |-  C   ande 1

4.  (All x)(P(x)/\C)    |-  (All x)P(x) aai 2

5.  (All x)(P(x)/\C)    |-  C   lai 3

6.  (All x)(P(x)/\C)    |-  (All x)P(x)/\C  andi 4,5

第三十五关

Premise: (All x)P(x)/\C

Conclusion: (All x)(P(x)/\C)

1.  (All x)P(x)/\C  |-  (All x)P(x)/\C  prem

2.  (All x)P(x)/\C  |-  (All x)P(x) ande 1

3.  (All x)P(x)/\C  |-  C   ande 1

4.  (All x)P(x)/\C  |-  P(b)    ae 2 (b/x)

5.  (All x)P(x)/\C  |-  P(b)/\C andi 3,4

6.  (All x)P(x)/\C  |-  (All x)(P(x)/\C)    ai 5 

第三十六关

Premise: ~(All x)P(x)

Conclusion: (Exist x)(~P(x))

1.  ~(All x)P(x),~(Exist x)(~P(x))  |-  ~(All x)P(x)    prem

2.  ~(All x)P(x),~(Exist x)(~P(x)),~P(b)    |-  ~(Exist x)(~P(x))   prem

3.  ~(All x)P(x),~(Exist x)(~P(x)),~P(b)    |-  ~P(b)   prem

4.  ~(All x)P(x),~(Exist x)(~P(x)),~P(b)    |-  (Exist x)(~P(x))    ei 3 (b/x)

5.  ~(All x)P(x),~(Exist x)(~P(x))  |-  ~~P(b)  ni 2,4

6.  ~(All x)P(x),~(Exist x)(~P(x))  |-  P(b)    nne 5

7.  ~(All x)P(x),~(Exist x)(~P(x))  |-  (All x)P(x) ai 6

8.  ~(All x)P(x)    |-  ~~(Exist x)(~P(x))  ni 1,7

9.  ~(All x)P(x)    |-  (Exist x)(~P(x))    nne 8

第三十七关

Premise: (Exist x)(~P(x))

Conclusion: ~(All x)P(x)

1.  ~P(b),(All x)P(x)   |-  ~P(b)   prem

2.  ~P(b),(All x)P(x)   |-  (All x)P(x) prem

3.  ~P(b),(All x)P(x)   |-  P(b)    ae 2 (b/x)

4.  ~P(b)   |-  ~(All x)P(x)    ni 1,3

5.  (Exist x)(~P(x))    |-  ~(All x)P(x)    lei 4

第三十八关

Premise: (Exist x)(P(x)/\Q(x))

Conclusion: (Exist x)P(x)/\(Exist x)Q(x)

1.  P(x)/\Q(x)  |-  P(x)/\Q(x)  prem

2.  P(x)/\Q(x)  |-  P(x)    ande 1

3.  P(x)/\Q(x)  |-  Q(x)    ande 1

4.  (Exist x)(P(x)/\Q(x))   |-  (Exist x)P(x)   eei 2

5.  (Exist x)(P(x)/\Q(x))   |-  (Exist x)Q(x)   eei 3

6.  (Exist x)(P(x)/\Q(x))   |-  (Exist x)P(x)/\(Exist x)Q(x)    andi 4,5 

初笔于2023年4月3日。 

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