# Logic/Logic patterns

A: a, b, 0, 1, +, -, ., /, =, =/=, ~, ().

B: ++ = +, +- = +-, -- = -, +.+ = +, +/+ = +, -.- = +, -/- = +, +.- = -, +/- = -, -/+ = -

C: + = + , - = -, . = ., / = /, ~ = ~, = = =, =/= = =/=, () = ()

D: + =/= -, . =/= /, ~ =/= -, = =/= =/=

E: =

1. a => a <=> a = a

2. ~a => ~a <=> ~a = ~a

3. a =/=> ~a <=> a =/= ~a

4. a ^ ɸ => a <=> a + 0 = a

5. a ^ ~ɸ => a <=> a - 0 = a

6. ~a ^ ɸ => ~a <=> ~a + 0 = ~a

7. ~a ^ ~ɸ => ~a <=> ~a - 0 = ~a

8. a v 1 => a <=> a.1 = a

9. a < 1 => a <=> a/1 = a

10. ~a v 1 => ~a <=> ~a.1 = ~a

11. ~a < 1 => ~a <=> ~a/1 = ~a

12. a v ɸ => ɸ <=> a.0 = 0

13. a < ɸ => ɸ <=> a/0 = 0

14. ~a v ɸ => ɸ <=> ~a.0 = 0

15. ~a < ɸ => ɸ <=> ~a/0 = 0

16. a ^ 1 => 1 ^ a <=> a + 1 = 1 + a

17. a ^ ~1 => Q ^ a <=> a - 1 = -1 + a

18. ~a ^ 1 => 1 ^ ~a <=> ~a + 1 = 1 + ~a

19. ~a ^ ~1 => Q ^ ~a <=> ~a - 1 = -1 + ~a

20. ɸ ^ a => a <=> 0 + a = a

21. ɸ ^ ~a => ~a <=> 0 - a = -a

22. ɸ ^ ~(~a) => !a <=> 0 - ~a = - ~a

23. a ^ a => a ^ a <=> a + a = a + a

24. a v a => a <=> a . a = a

25. a ^ ~a => e <=> a - a = 0

26. a < a => 1 <=> a / a = 1

27. a v ~a => Q <=> a . ~a = 0

28. a ^ ~(~a) => a ^ !a <=> a - ~a = 1

29. a < ~a => Q <=> a / ~a = -1

30. a v b => b v a <=> a . b = b . a

31. a ^ b => b ^ a <=> a + b = b + a

32. (a ^ b) ^ c => a ^ (b ^ c) <=> (a + b) + c = a + (b + c)

33. (a v b) v c => a v (b v c) <=> (a . b) . c = a . (b . c)

34. a ^ (b v c) => (a ^ b) v (a ^ c) <=> a + (b . c) = (a + b) . (a + c)

35. a v (b ^ c) => (a v b) ^ (a v c) <=> a . (b + c) = (a . b) + (a . c)

36. ~(a ^ b) => ~a ^ ~b <=> ~(a + b) = ~a + ~b

37. ~(a v b) => ~a v ~b <=> ~(a . b) = ~a . ~b

38. a < b => a < b <=> a / b = a / b

39. a ^ ~b => ~b ^ a <=> a - b = -b + a

40. (a ^ ~b) ^ ~c => a ^ (~b ^ ~c) <=> (a - b) - c = a (-b - c)

41.i.(a < b) < c => a < (b v c) <=> (a / b) / c = a / (b . c)

41.ii. a < (b < c) => (a v c) < b <=> a / (b / c) = (a . c) / b

42. a ^ ~(b < c) => (a ^ ~b) < (a ^ ~c) <=> a - (b / c) = (a - b) / (a - c)

43. a < (b ^ ~c) => (a < b) ^ ~(a < c) <=> a / (b - c) = (a / b) - (a / c)

44. ~(a ^ ~b) => ~a ^ ~(~b) <=> ~(a - b) = ~a - ~b

45. ~(a < b) => ~a < ~b <=> ~(a / b) = ~a / ~b

46. ʘ => ʒ ^ e

47. ψa => a ^ ~a ^ !a

48. Эa => a v ~a v !a

49. ʒ => e

50. ɸ => ʒ ^ ~e

51. 1 => a < a

52. e => a ^ ~a

53. Q => ~ɸ ^ ~1

54. !a => ~(~a)

55. ʘ => ʒ ^ e < (Q ^ !e) v [(ɸ ^ 1) ^ (ψ ^ Э)]

zi': 0 + 0 = 0 <=> ɸ ^ ɸ => ɸ

zi": 0 - 0 = 0 <=> ɸ ^ ~ɸ => Q

zi"': 0 . 0 = 0 <=> ɸ v ɸ => ɸ

zi"": 0 / 0 = 0 <=> ɸ < ɸ => ∞

zii':0 . 1 = 0 <=> ɸ v 1 => ɸ

zii": 1 . 0 = 0 <=> 1 v ɸ => ɸ

zii"': 0 / 1 = 0 <=> ɸ < 1 => ɸ

zii"": 1 / 0 = 0 <=> 1 < ɸ => ɸ

ziii': 0 . a = 0 <=> ɸ v a => ɸ

ziii": a . 0 = 0 <=> a v ɸ => ɸ

ziii"': 0 / a = 0 <=> ɸ < a => ɸ

ziii"": a / 0 = 0 <=> a < ɸ => ɸ

ziv': 0 . ~a = 0 <=> ɸ v ~a => ɸ

ziv": ~a . 0 = 0 <=> ~a v ɸ => ɸ

ziv"': 0 / ~a = 0 <=> ɸ < ~a => ɸ

ziv"": ~a / 0 = 0 <=> ~a < ɸ => ɸ

zv': 1 - 1 = 0 <=> 1 ^ ~1 => Q

zv": a - a = 0 <=> a ^ ~a => e

zv"': ~a - ~a = 0 <=> ~a ^ ~(~a) => !ɸ

zv"": a + ~a = 0 <=> a ^ ~a => e

G1. ЭX < ЭY

G2. ЭX < Э~Y

G3. ЭY < ЭX

G4. ЭY < Э~X

G5. Э~X < ЭY

G6. Э~Y < ЭX

G7. Э~X < Э~Y

G8. Э~Y < Э~X

G9. [~]X < ЭY

G10. [~]Y < ЭX

G11. [~]Y < Э~X

G12. [~]~X < ЭY

G13. [~]X < Э~Y

G14. [~]~Y < ЭX

G15. [~]~X < ~Y

G16. [~]~Y < ~X

P1. (ЭX < ЭY) ^ ([~]Y < Э~X) => (ЭX < ψY)

P2. (ЭX < Э~Y) ^ ([~]X < ЭY) => (ψX < Э~Y)

P3. (ЭX < ЭY) ^ (ЭY < ЭX) => (ЭX < ЭY)

P4. (Э~X < ЭY) ^ (ЭY < Э~X) => (Э~X < ЭY)

One.

1. (ψB < ~A) => (ψA < ~B)

2. (ЭA < ψC) ^ (ЭC < ЭB) =/=> [(ψB < ~A) <= (ЭC < ЭB)]

3. (ψB < ЭA) => (ЭA < ψB)

4. (ψA < ~B) => (ψB < ~A)

5. (ψB < ЭA)

6. (ЭB < ЭA) => (ЭA < ЭB)

7. (ψA < ~B) => (ψB < ~A)

8. (ЭB < ~A) => [(ЭA < ЭB) v (ψA < ~B)]

Two.

1. (ЭA < ψB) ^ (ЭB < ψC) => (ЭA < ψC)

2. (ψA < ~B) ^ (ЭB < ψC) => (ψA < ~C)

Three.

1. (ψB < ЭA) ^ (ЭC < ЭB) => (ЭC < ЭA)

2. (ψB < ~A) ^ (ЭC < ЭB) => (ЭC < ~A)

Four.

1. (ψA < ~B) ^ (ЭA < ψC)

2. (ψA < ~B) <=> (ψB < ~A) <= [(ψ < ~) <=> (~ < ψ)]

3. (ЭA < ψC)

4. (ψB < ~C)

5. (ЭA < ψB) ^ (ψA < ~C) => (ψB < ~C)

6. (ψA < ~C) => (ψC < ~A)

7. (ЭA < ψB)

8. (ψC < ~B)

9. (ψC < ~B) => (ψB < ~C)

Five.

1. (ψA < ~B) ^ (ЭA < ЭC) => (ЭC < ~B)

2. [(ψA < ~B) => (ψB < ~A)] <= [(ψ < ~) <=> (~ < ψ)]

3. (ЭA < ЭC)

4. (ЭC < ~B)

5. (ЭA < ψB) ^ (ψA < ~C) => (ψB < ~C)

6. (ЭB < ψC) ^ (ЭA < ψB) => (ЭA < ψC)

7. (ψA < ~C)

8. (ЭA < ψB) ^ (ЭC < ~A) => (ЭC < ~B)

Six.

1. (ЭA < ψB) ^ (ЭA < ЭC)

2. (ЭB < ψC) v (ψB < ~C)

3. (ψA < ~C) ^ (ЭA < ~B)

4. (ЭB < ψC) v (ψB < ~C)

Seven.

1. (ЭA < ψC) ^ (ЭB < ψC) => (ЭA < ЭB)

2. [(ЭB < ψC) <=> (ψC < ЭB)] <= [(Э < ψ) <=> (ψ < Э)]

3. (ЭA < ψC) ^ (ψC < ЭB) => (ЭA < ЭB)

Eight.

1. [(ЭA < ψC) ^ (ЭB < ψC)] ^ (ЭC < ψD)

2. [(ЭA < ψD) ^ (ЭB < ψD)] => (ЭA < ЭB)

3. (ЭB < ψC) ^ (ψA < ~C) => (ЭB < ~A)

Nine.

1. (ЭB < ψC) ^ (ЭA < ЭC) => (ЭA < ЭB)

2. [(ЭA < ЭC) <=> (ЭC < ЭA)] <= [(Э < Э) <=> (Э < Э)]

3. (ЭB < ψC) ^ (ЭC < ЭA) => (ЭB < ЭA)]

4. (ЭA < ЭB)]

Ten.

1. (ЭB < ЭC) ^ (ЭA < ψC) => (ЭA < ЭB)

2. (ЭB < ψC) ^ (ЭA < ~C) => (ЭA < ~B)

3. (ЭA < ψB) ^ (ЭB < ψC) => (ЭA < ψC)

4. (ЭA < ~C)

Eleven.

1. (ψA < ~C) ^ (ЭB < ЭC) => (ЭA < ~B)

Twelve.

1. [(ЭA < ψB) v (ЭA < ЭB)] ^ (ψB < ~C)

2. [(ψB < ЭA) v (ЭB < ЭA)] ^ (ψC < ~B)

3. (ЭA < ~C)

Thirteen.

1. (ЭA < ψC) ^ (ЭB < ψC) => (ЭA < ЭB)

2. (ψA < ~B) ^ (ЭB < ψC) => (ψA < ~C)

3. (ЭA < ψC)

Fourteen.

1. (ЭA < ψB) ^ (ЭB < ЭC) => (ЭA < ЭC)

2. (ψA < ~C) ^ (ЭA < ψB) => (ψB < ~C)

3. (ψA < ~B) ^ (ЭB < ЭC) => (ЭC < ~A)

4. (ЭA < ψC) ^ (ψA < ~B) => (ψB < ~C)

Fifteen.

1. (ЭA < ψB) ^ (ЭB < ЭC)

2. (ЭA < ЭC)

3. (ЭB < ЭC) ^ (ЭA < ψB)

Sixteen.

1. (ψB < ~A) ^ (ЭA < ψC)

2. [(ψB < ~A) => (ψA < ~B)] <= [(ψ < ~) => (~ < ψ)]

3. (ЭA < ψC)

4. (ψB < ~C)

5. (ψC < ЭA)

Seventeen.

1. (ψA < ~C) => (ψC < ~A)

2. (ЭA < ψB)

3. (ψC < ~B)

4. (ψB < ~C)

Eighteen.

1. (ψC < ~A)

2. (ψB < ~C) => (ψC < ~B)

3. (ЭB < ЭA) <= (ЭA < ψB)

4.(ψC < ~A)

Nineteen.

1. (ψB < ~A) ^ (ЭC < ЭA)

2. (ψB < ~A) => (ψA < ~B)

3. (ЭC < ЭA)

4. (ЭB < ~C)

Twenty.

1. (ЭA < ψB) ^(ЭC < ~A) => (ЭC < ~B)

Twenty One.

1. (ЭA < ψC) ^ (ЭB < ψC)

2. [(ЭB < ψC) => (ЭC < ЭB)] <= [(Э < ψ) => (Э < Э)]

3. (ЭA < ψC) ^ (ЭC < ЭB) => (ЭA < ЭB)

4. (ЭC < ЭB)

Twenty Two.

1. (ЭC < ЭA) => (ЭA < ЭC)

2. (ЭB < ψC) => (ЭB < ЭA)

Twenty Three.

1. (ψA < ~C) ^ (ЭB < ЭC)

2. (ЭB < ЭC) ^ (ψA < ~C) => (ЭA < ~B)

3. (ЭB < ЭC)

Twenty Four.

1. (ЭB < ЭC) ^ (ψA < ~C)

2. (ЭB < ЭC) => (ЭC < ЭB)

3. (ψA < ~C) ^ (ЭB < ЭC) => (ЭA < ~B)

Twenty Five.

1. (ЭB < ψC) ^ (ЭA < ЭC) => (ЭB < ЭA)

2. (ψA < ~C) ^ (ЭB < ЭC) => (ЭB < ~A)

Twenty Six.

1. [(ЭA < ψB) v (ЭA < ЭB)] ^ (ЭB < ψC)

2. (ЭA < ψC) v (ЭA < ЭC)

3. [(ψA < ~B) v (ЭA < ~B)] ^ (ЭB < ψC)

4. (ψA < ~C) v (ЭA < ~C)

5. {[(ЭA < ψB) v (ЭA < ЭB)] ^ (ЭB < ЭC)} => [(ЭA < ψC) v (ЭA < ЭC)]

Twenty Seven.

1. {[(ψA < ~B) v (ЭA < ~B)] ^ (ЭB < ЭC)} => [(ψA < ~C) v (ЭA < ~C)]

Twenty Eight.

1. [(ЭA < ψB) v (ЭA < ЭB)] ^ (ЭB < ψC)

2. {(ЭB < ψC) ^ [(ЭA < ψB) v (ЭA < ЭB)]} => [(ЭA < ψC) v (ЭA < ЭC)]

Twenty Nine.

1. {[(ψA < ~B) v (ЭA < ~B)] ^ (ЭB < ψC )} => [(ψA < ~C) v (ЭA < ~C)]

Thirty.

1. A < B

2. [(A <= B) v (A => B v ~B)] ^ (~A => ~B)

3. (A v ~A) ^ ~B

4. (A v ~A) => (A v ~A)

5. ~B => ~B

6. (A v ~A) ^ ~B

7. [(A ^ ~B) v (~A ^ ~B)] => [(A < A) ^ (A => A)]

8. (A => A) <=> (A < A)

Thirty One.

1. (A v ~A) ^ (A < B) => [(A => B) v (~A => ~B)]

2. A <= B

3. {[(A => B v ~B) v (~A => ~B)] ^ (A v ~A)} => (B v ~B)

4. (~B v ~B) => [(~B v ~B) ^ (B v ~B)]

Thirty Two.

1. (~B v ~B) ^ ~(B v B) ^ ~(ψB v ЭB)

2. (ЭB < ψC)

3. (ψA < ~C) ^ (ЭB < ψC) => [(ЭA < ~B) v (ψA < ~B)]

4. (ЭA < ψB) v (ЭA < ЭB)

5. (ЭA < ψC) v (ЭA < ЭC)

6. (ψA < ~C) <=/=> [(ЭA < ψC) v (ЭA < ЭC)]

Thirty Three.

1. B < C

2. {(ЭB < ψC) ^ [(ЭA < ψB) v (ЭA < ЭB)]} => [(ЭA < ψC) v (ЭA < ЭC)]

3. (ψA < ~C) v (ЭA < ~C)

Thirty Four.

1. (ψA < ~B) ^ [(ЭB < ψC) v (ЭB < ЭC)]

2. (ЭA < ~C) v (ψA < ~C)

3. (ψA < ~C) ^ [(ЭB < ψC) v (ЭB < ЭC)]

4. (ЭA < ЭB)

5. (ψA < ~B) <=/=> (ЭA < ЭB)

6. (ψA < ~C)

Thirty Five.

1. (ЭA < ψB) ^ [(ψB < ~C) v (ЭB < ~C)]

2. (ЭA < ~C) v (ЭA < ~C)

3. (ЭB < ψC) v (ЭB < ЭC)

4. {(ЭA < ψB) ^ [(ЭB < ψC) v (ЭB < ЭC)]} => [(ЭA < ψC) v (ЭA < ЭC)]

Thirty Six.

1. (ψA < ~B) ^ [(ЭB < ~C) v (ψB < ~C)]

2. [(ЭA < ψC) v (ЭA < ЭC)] v [(ЭA < ~C) v (ψA < ~C)]

3. {(ЭA < ψB) ^ [(ЭB < ~C) v (ψB < ~C)]} => [(ЭA < ~C) v (ψA < ~C)]

Thirty Seven.

1. (ψA < ~B) ^ [(ЭB < ~C) v (ψB < ~C)]

2. [(ЭA < ψC) v (ЭA < ЭC)] v [(ЭA < ~C) v (ψA < ~C)]

3. {[(ЭB < ψC) v (ЭB < ЭC)] ^ (ψA < ~B)} => [(ЭA < ~C) v (ψA < ~C)]

Thirty Eight.

1. (ЭA < ψB) ^ [(ЭB < ψC) v (ЭB < ЭC)]

2. [(ЭA < ψC) v (ЭA < ЭC)]

Thirty Nine.

1. [(ЭA < ψB) v (ЭA < ЭB)] ^ (ЭB < ψC)

2. [(ЭA < ЭC) v (ЭA < ψC)] =/=> (ЭA < ψC)]

Forty.

1. (ψB < ~A) ^ [(ЭB < ЭC) v (ЭB < ψC)]

2. (ψA < ~C)

3. [(ЭA < ЭC) v (ЭA < ψC)]

4. (ψB < ~A)

5. (ψB < ~A) => (ψA < ~B)

6. [(ЭA < ЭC) v (ЭA < ψC)]

7. [(ЭB < ~C) v (ψB < ~C)]

8. [(ЭB < ЭC) v (ЭB < ψC)]

Forty One.

1. (ψB < ~A) ^ [(ЭB < ЭC) v (ЭB < ψC)]

2. (ЭA < ~C)

3. (ЭA < ψC) ^ [(ψB < ~A) => (ψA < ~B)]

4. (ЭA < ψC) => (ψC < ~B)

5. [(ЭB < ЭC) v (ЭB < ψC)]

Forty Two.

1. (ψA < ~B) ^ (ЭA < ψC)

2. (ψA < ~B) => (ψB < ~A)

3. (ЭA < ψC)

4. (ψB < ~A) ^ (ЭA < ψC) => (ψB < ~C)

Forty Three.

1. (ψB < ~A) ^ [(ЭA < ψC) v (ЭA < ЭC)]

2. (ψB < ~A) => (ψA < ~B)

3. [(ЭA < ψC) v (ЭA < ЭC)]

4. {(ψA < ~B) ^ [(ЭA < ψC) v (ЭA < ЭC)]} => [(ψC < ~B) v (ЭC < ~B)]

5. (ψC < ~B)

6. (ЭC < ЭB)

7. (ψA < ~B) ^ (ЭC < ЭB) => (ЭA < ~C)

8. [(ЭA < ψC) v (ЭA < ЭC)]

Forty Four.

1. (ψB < ~A) ^ [(ψC < ~A) v (ЭC < ~A) v (ЭC < ЭA)]

2. (ψB < ~A) => (ψA < ~B)

3. (ЭA < ψC)

4. (ψA < ~B) ^ (ЭA < ψC) => (ψB < ~C)

Forty Five.

1. [(ЭA < ψC) v (ЭA < ЭC)] ^ [(ЭB < ψC) v (ЭB < ЭC)]

2. [(ЭB < ψC) v (ЭB < ЭC)] => [ψC < ЭB) v (ЭC < ЭB)]

3. [(ЭA < ψC) v (ЭA < ЭC)] ^ [ψC < ЭB) v (ЭC < ЭB)]

4. (ЭA < ЭB) v (ψA < ~B)

Forty Six.

1. [(ψC < ~A) v (ЭC < ЭA)] ^ [(ЭB < ψC) v (ЭB < ЭC)]

2. (ЭB < ~A) v (ЭB < ЭA)

Forty Seven.

1. [(ψC < ~A) v (ЭC < ~A)] ^ [(ψC < ~B) v (ЭC < ~B)]

2. [(ψB < ~A) v (ЭB < ~A)] v [(ЭB < ψA) v (ЭB < ЭA)]

3. {[(ЭA < ψC) v (ЭA < ЭC)] ^ [(ЭB < ψC) v (ЭB < ЭC)]} => [(ЭB < ψA) v (ЭB < ЭA)]

Forty Eight.

1. [(ЭA < ψC) v (ЭA < ЭC)] ^ (ЭB < ЭC)

2. (ЭB < ЭC) => (ЭC < ЭB)]

3. {[(ЭA < ψC) v (ЭA < ЭC)] ^ (ЭC < ЭB)} => [(ЭA < ЭB) v (ЭA < ~B)]

Forty Nine.

1. (ЭA < ψC) ^ [(ЭB < ψC) v (ψB < ~C)]

2. [(ЭB < ψC) v (ψB < ~C)] => [(ψC < ЭB) v (ψC < ~B)]

3. {(ЭA < ψC) ^ [(ψC < ЭB) v (ψC < ~B)]} => [(ЭA < ЭB) v (ЭA < ~B)]

Fifty.

1. (ЭB < ψC) ^ [(ЭA < ЭC) v (ЭA < ~C)]

2. [(ЭA < ЭB) v (ЭA < ~B)]

3. (ЭA < ψB) ^ (ЭB < ψC)

4. (ЭA < ψC)

5. [(ЭA < ЭC) v (ЭA < ~C)]

Fifty One.

1. (ЭA < ψC) ^ [(ЭB < ψC) v (ЭB < ЭC)]

2. (ЭA < ψC) ^ [(ЭC < ЭB) v (ψC < ЭB)]

3. [(ЭB < ψA) v (ЭB < ЭA)]

Fifty Two.

1. A=>B, C=>A, D=/=>A.

2. E=>F, G=>E, H=/=>E.

3. C <=> F => ЭA < ψE.

4. (ЭF < ψE) ^ (ЭA < ψC) => (ЭA < ψE)

5. [(C <=> G) => (ЭA < ЭE)] <= [(C=>A) ^ (G=>E)]

6. (F <=> D) => (ψA < ~E)

7.{[(ψA < ~E) => (ψE < ~A)] ^ (F <=> D)} => [(ψA < ~F) ^ (ЭF < ψE)].

Fifty Three.

1. (B <=> H) => (ψA < ~E).

2. (ЭB < ψA) ^ (ψB < ~E).

3. (B <=> H) ^ (ψH < ~E).

Fifty Four.

1. (D <=> G) => [(ψA v ЭA) < ~E.]

2. (ψA < ~G) <= (ψA < ~D).

3. ψG < ЭE.

4. (ψA v ЭA) < ~E.

Fifty Five.

1. (B <=> G).

2. (ЭE < ψA) <= {[(ψB < ЭA) ^ (ψE < ЭB)] ^ (B <=> G)}.

3. (ЭA < ψE) v (ЭA < ЭE).

4. (ЭA < ЭE) <= (ψ => Э)

5. (ЭC <=> ЭF) => (ЭA < ψE)

Fifty Six.

1. (C <=> G) => (ЭA <=> ЭE)

2. (ЭD <=> ЭF) => (ψA < ~E)

3. [(ψA < ~F) <= {(ψF < ~A)] ^ (ЭF < ψE)}

4. (ψD < ~A) ^ (ЭD < ψE)

5. (ЭD <=> ЭG) => (ψA < ~E)

6. (ψA < ~G) ^ (ЭE < ψG)

S1. (ЭC < ψB) ^ (ЭB < ψA) => (ЭC < ψA)

S2. (ψC < ~B) ^ (ЭB < ψA) => (ψC < ~A)

S3. (ЭC < ψB) ^ (ЭB < ЭA) => (ЭC < ЭA)

S4. (ψC < ~B) ^ (ЭB < ЭA) => (~C < ЭA)

S5. (ψB < ~C) ^ (ЭB < ψA) => (ψC < ~A)

S6. (ЭB < ψC) ^ (ψB < ~A) => (ψC < ~A)

S7. (ψB < ~C) ^ (ЭB < ЭA) => (~C < ЭA)

S8. (ЭB < ψC) ^ (~B < ЭA) => (~C < ЭA)

S9. (ЭC < ψB) ^ (ЭA < ψB) => (ЭC < ЭA)

S10. (ψC < ~B) ^ (ЭA < ψB) => (~C < ЭA)

S11. (ЭC < ЭB) ^ (ЭA < ψB) => (ЭC < ЭA)

S12. (ЭC < ψB) ^ (ЭA < ЭB) => (ЭC < ЭA)

S13. (~C < ЭB) ^ (ЭA < ψB) => (~C < ЭA)

S14. (ψC < ~B) ^ (ЭA < ЭB) => (~C < ЭA)

S15.(ЭB < ψC) ^ (ЭA < ψB) => (ψC < ЭA)

S16.(ЭB < ψC) ^ (ψA < ~B) => (~C < ψA)

S17.(ЭB < ЭC) ^ (ЭA < ψB) => (ЭC < ЭA)

S18.(ψB < ~C) ^ (ЭA < ψB) => (~C < ЭA)

S19.(ψB < ~C) ^ (ЭA < ЭB) => (~C < ЭA)

1. (ЭC < ψA)

1.1 (ЭC < ψB) ^ (ЭB < ψA)

2. (ψC < ~A)

2.1 (ψC < ~B) ^ (ЭB < ψA)

2.2 (ψB < ~C) ^ (ЭB < ψA)

2.3 (ЭB < ψC) ^ (ψB < ~A)

3. (ЭC < ЭA)

3.1 (ЭC < ψB) ^ (ЭB < ЭA)

3.2 (ЭC < ψB) ^ (ЭA < ψB)

3.3 (ЭC < ψB) ^ (ЭA < ЭB)

3.4 (ЭC < ЭB) ^ (ЭA < ψB)

3.5 (ЭB < ψC) ^ (ЭA < ψB)

3.6 (ЭB < ЭC) ^ (ЭA < ψB)

4.(~C < ЭA)

4.1 (ψC < ~B) ^ (ЭB < ЭA)

4.2 (ψB < ~C) ^ (ЭB < ЭA)

4.3 (ЭB < ψC) ^ (~B < ЭA)

4.4 (ψC < ~B) ^ (ЭA < ψB)

4.5 (~C < ЭB) ^ (ЭA < ψB)

4.6 (ψC < ~B) ^ (ЭA < ЭB)

4.7 (ЭB < ψC) ^ (ψA < ~B)

4.8 (ψB < ~C) ^ (ЭA < ψB)

4.9 (ψB < ~C) ^ (ЭA < ЭB)

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