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Harjoitustehtävät. Mereologia ja sen soveltaminen Laskuharjoitus 2 M. Keinänen. HT 6. A1.  x  y ((x ○ y) →  z  w (( w < z) ↔ ((w < x) Λ (w < y)))) ├  x  y ( ¬ ( x < y) → (  z )((z < x) Λ ¬ ( z ○ x))) ¬ ( x < y) Apupremissi ( x ○ y) ν ¬ (x ○ y) Tautologia

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### Harjoitustehtävät

Mereologia ja sen soveltaminen

M. Keinänen

A1. xy ((x ○ y) → z w ((w < z) ↔ ((w < x) Λ (w < y))))

├ xy (¬ (x < y) → (z)((z < x) Λ¬ (z ○ x)))

• ¬ (x < y) Apupremissi

• (x ○ y) ν¬ (x ○ y) Tautologia

• ¬(x ○ y) Apupremissi

• x < x <, teoreema, US x/x

• (z)(z < x) Λ¬(z ○ y) EG, lausel. 3, 4

• ¬(x ○ y) → (z)(z < x) Λ¬(z ○ y) CP 3,..5

• (x ○ y) Apupremissi

• (x ○ y) → z w ((w < z) ↔ ((w < x) Λ (w < y))) US, A1

• z w ((w < z) ↔ ((w < x) Λ (w < y))) MP 7,8

• w ((w < a) ↔ ((w < x) Λ (w < y)) ES 9, a/z, a=axy

• (a < a) ↔ ((a < x) Λ (a < y)) US 10, a/w

• (a < x) Λ (a < y)

• (x < a) Vastaoletus

• (x = a) <, teoreema 12,13

• (x < y) 14 IK, RR

• ¬ (x < y) Λ (x < y) RR

• ¬(x < a) CP 13, …16

• (¬(x < a) Λ (a < x)) → (a « x) US x/x, teoreema

• (a « x) MP

• (a « x) →(z)((z « x) Λ¬ (z ○ a) US a/x, y/y, PA 11

• (z)((z « x) Λ¬ (z ○ a) MP

• (b « x) Λ¬ (b ○ a) ES 21 b/z

• (t < a) ↔ ((t < x) Λ (t < y))

23. (b < x) Λ¬ (b ○ a) Lauselogiikka, Määr. 22

24. ( w)((w < b) Λ(w < y)) vastaoletus

25. (c < b) Λ(c < y) ES 24 c/w

26. (c < x) Λ(c < y) Määr., Lauselogiikka 23, 26

27. ((c < a) ↔ ((c < x) Λ (c < y)) US 10 c/w

28. (c < a) Lauselogiikka 26, 27

29. ¬ (b ○ a) Lauselogiikka 23

30. x ¬ ((x < b) Λ (x < a)) 29. Määr.

31. ¬ ((c < b) Λ (c < a)) US 30, c/b RR 25,28

32. ¬( w)((w < b) Λ(w < y)) 24, 31…. ¬24

33. ¬ (b ○ y) Määr. 32

34. (b < x) Λ¬ (b ○ y) 23, 33 lauselogiikka

• ( z) ((z < x) Λ¬ (z ○ y)) 34, EG, z/b

• (x ○ y) → ( z) ((z < x) Λ¬ (z ○ y)) CP, 7-35

• ( z) ((z < x) Λ¬ (z ○ y)) CP, 2,6,36

• ¬ (x < y) → ( z) ((z < x) Λ¬ (z ○ y)) CP, 1-37

• xy (¬ (x < y) → (z)((z < x) Λ¬ (z ○ x))) UG 38 x/x, y/y