03 - Predicate logic
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The document discusses various concepts related to predicate logic, including boolean algebra, propositional formulas, and the truth values of statements using logical connectives. It presents a problem involving three characters—Stirlitz, Mueller, and Eismann—where it is deduced that Eismann is not a Russian spy based on the given conditions and logical reasoning. Additionally, the text touches on properties of natural numbers and quantifiers in mathematical logic.
![Example
Given: an array a[0..N-1] of length N≥3
Find: i, j, k such that:
i is the index of the greatest element of a;
j is the index of the 2nd greatest element of a;
k is the index of the 3rd greatest element of a.](https://image.slidesharecdn.com/fgdi-03-predicatedlogic-101103140601-phpapp01/85/03-Predicate-logic-73-320.jpg)
![Example
Given: an array a[0..N-1] of length N≥3
Find: i, j, k such that:
i is the index of the greatest element of a;
j is the index of the 2nd greatest element of a;
k is the index of the 3rd greatest element of a.
∃ i,j,k: (0≤i,j,k<N)∧(i≠j)∧(i≠k)∧(j≠k):
( (ai≥aj≥ak) ∧
(∀m: (0≤m<N)∧(m≠i)∧(m≠j)∧(m≠k): ak≥am) )](https://image.slidesharecdn.com/fgdi-03-predicatedlogic-101103140601-phpapp01/85/03-Predicate-logic-74-320.jpg)
![Min x: U(x): T(x)
gives the “smallest value of terms T(x)
for all objects x from the universe U“
Min x: False: T(x) is +∞.
( m = (Min i: 0≤i<N: a[i]) )](https://image.slidesharecdn.com/fgdi-03-predicatedlogic-101103140601-phpapp01/85/03-Predicate-logic-92-320.jpg)
![Min x: U(x): T(x)
gives the “smallest value of terms T(x)
for all objects x from the universe U“
Min x: False: T(x) is +∞.
( m = (Min i: 0≤i<N: a[i]) )
((∃ i: 0≤i<N: m=a[i]) ∧ (∀ i: 0≤i<N: m≤a[i]))](https://image.slidesharecdn.com/fgdi-03-predicatedlogic-101103140601-phpapp01/85/03-Predicate-logic-93-320.jpg)
03 - Predicate logic
- 2. Boolean algebra Propositional logic 0, 1 True, False boolean variables: a∈{0, 1} atomic formulas: p∈{True,False} boolean operators: ¬ ∧ ∨ logical connectives: ¬ ∧ ∨ boolean functions: 0, 1 boolean variables if a is a boolean function, then ¬a is a boolean function if a and b are boolean functions, then a∧b, a∨b, a b, a b are boolean functions propositional formulas (propositions): True and False atomic formulas if a is a propositional formula, then ¬a is a propositional formula if a and b are propositional formulas, then a∧b, a∨b, a b, a b are propositional formulas truth value of a boolean function (truth tables) truth value of a propositional formula (truth tables)
- 3. Boolean algebra Propositional logic 0, 1 True, False boolean variables: a∈{0, 1} atomic formulas: p∈{True,False} boolean operators: ¬ ∧ ∨ logical connectives: ¬ ∧ ∨ boolean functions: 0, 1 boolean variables if a is a boolean function, then ¬a is a boolean function if a and b are boolean functions, then a∧b, a∨b, a b, a b are boolean functions propositional formulas (propositions): True and False atomic formulas if a is a propositional formula, then ¬a is a propositional formula if a and b are propositional formulas, then a∧b, a∨b, a b, a b are propositional formulas truth value of a boolean function (truth tables) truth value of a propositional formula (truth tables)
- 4. The Russian spy problem There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. Moreover, every Russian must be a spy. When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. Establish that Eismann is not a Russian spy.
- 5. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. Moreover, every Russian must be a spy. When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. Establish that Eismann is not a Russian spy.
- 6. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. Moreover, every Russian must be a spy. When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. Establish that Eismann is not a Russian spy. X ∈ {R, G, S} (denoting Russian, German, Spy) Y ∈ {S,M,E} (denoting Stirlitz, Müller, Eismann) SE : Eismann is a Spy RS : Stirlitz is Russian
- 7. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. (RS ∧ GM ∧ GE) ∨ (GS ∧ RM ∧ GE) ∨ (GS ∧ GM ∧ RE). (RS ¬GS) ∧ (RM ¬GM) ∧ (RE ¬GE). Moreover, every Russian must be a spy. (RS SS) ∧ (RM SM) ∧ (RE SE). When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. RS GM. Establish that Eismann is not a Russian spy. RE ∧ SE.
- 8. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. (RS ∧ GM ∧ GE) ∨ (GS ∧ RM ∧ GE) ∨ (GS ∧ GM ∧ RE). (RS ¬GS) ∧ (RM ¬GM) ∧ (RE ¬GE). Moreover, every Russian must be a spy. (RS SS) ∧ (RM SM) ∧ (RE SE). When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. RS GM. Establish that Eismann is not a Russian spy. RE ∧ SE.
- 9. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. (RS ∧ GM ∧ GE) ∨ (GS ∧ RM ∧ GE) ∨ (GS ∧ GM ∧ RE). (RS ¬GS) ∧ (RM ¬GM) ∧ (RE ¬GE). Moreover, every Russian must be a spy. (RS SS) ∧ (RM SM) ∧ (RE SE). When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. RS GM. Establish that Eismann is not a Russian spy. RE ∧ SE.
- 10. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. (RS ∧ GM ∧ GE) ∨ (GS ∧ RM ∧ GE) ∨ (GS ∧ GM ∧ RE). (RS ¬GS) ∧ (RM ¬GM) ∧ (RE ¬GE). Moreover, every Russian must be a spy. (RS SS) ∧ (RM SM) ∧ (RE SE). When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. RS GM. Establish that Eismann is not a Russian spy. RE ∧ SE.
- 11. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. (RS ∧ GM ∧ GE) ∨ (GS ∧ RM ∧ GE) ∨ (GS ∧ GM ∧ RE). (RS ¬GS) ∧ (RM ¬GM) ∧ (RE ¬GE). Moreover, every Russian must be a spy. (RS SS) ∧ (RM SM) ∧ (RE SE). When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. RS GM. Establish that Eismann is not a Russian spy. RE ∧ SE.
- 12. There are three persons: Stirlitz, Mueller, and Eismann. It is known that exactly one of them is Russian, while the other two are Germans. (RS ∧ GM ∧ GE) ∨ (GS ∧ RM ∧ GE) ∨ (GS ∧ GM ∧ RE). (RS ¬GS) ∧ (RM ¬GM) ∧ (RE ¬GE). Moreover, every Russian must be a spy. (RS SS) ∧ (RM SM) ∧ (RE SE). When Stirlitz meets Mueller in a corridor, he makes the following joke: “you know, Mueller, you are as German as I am Russian”. It is known that Stirlitz always says the truth when he is joking. RS GM. Establish that Eismann is not a Russian spy. RE ∧ SE.
- 14. 2 is greater than 5
- 15. 2 is greater than 5 False
- 16. 2 is greater than 5 6 is greater than 5 False
- 17. 2 is greater than 5 6 is greater than 5 False True
- 18. 2 is greater than 5 6 is greater than 5 False True x is greater than 5
- 19. 2 is greater than 5 6 is greater than 5 False True x is greater than 5 ?
- 20. 2 is greater than 5 6 is greater than 5 False True x is greater than 5 ? predicate
- 21. x is greater than 5 2 is greater than 5 6 is greater than 5 False True predicate
- 22. x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5 False True predicate
- 23. x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True predicate
- 24. x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 predicate
- 25. x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 False predicate
- 26. x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 False for some natural numbers x, x is greater than 5 predicate
- 27. True x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 False for some natural numbers x, x is greater than 5 predicate
- 28. True x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 False for some natural numbers x, x is greater than 5 predicate quantifier
- 29. True x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 False for some natural numbers x, x is greater than 5 predicate quantifier quantifier
- 30. True x is greater than 5F(x) = 2 is greater than 5 6 is greater than 5F(6) = F(2) = False True for all natural numbers x, x is greater than 5 False for some natural numbers x, x is greater than 5 predicate quantifier quantifier universe
- 31. Existential quantifiers ∃ :Universal quantifier ∀ : ∀x: U(x): F(x) ∃x: U(x): F(x) ∀x:x∈N:x>5 ∃x:x∈N:x>5 for all natural numbers x, x is greater than 5 for some natural numbers x, x is greater than 5
- 32. Gottlob Frege 1848—1925 no appeal to intuition
- 33. Distributivity of ∀ ∧ ∀x: U(x): (P(x) ∧ Q(x)) Distributivity of ∃ ∨ ∃x: U(x): (P(x) ∨ Q(x)) Non-distributivity of ∀ ∨ (∀x: U(x): P(x)) ∨ (∀x: U(x): Q(x)) Non-distributivity of ∃ ∧ ∃x: U(x): (P(x) ∧ Q(x)) (∀x: U(x): P(x)) ∧ (∀ x: U(x): Q(x)) (∃x: U(x): P(x)) ∨ (∃ x: U(x): Q(x)) ∀x: U(x): (P(x) ∨ Q(x)) (∃x: U(x): P(x)) ∧ (∃x: U(x): Q(x))
- 34. Examples
- 35. ∀ i: 0≤i<N: ai=0 Examples
- 36. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero Examples
- 37. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai Examples
- 38. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing Examples
- 39. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj Examples
- 40. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing Examples
- 41. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj Examples
- 42. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal Examples
- 43. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 Examples
- 44. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a Examples
- 45. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 Examples
- 46. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero Examples
- 47. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai Examples
- 48. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing Examples
- 49. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj Examples
- 50. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj elements of array a are not monotonically increasing Examples
- 51. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai ≠ aj Examples
- 52. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai ≠ aj not all elements of array a are equal Examples
- 53. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai ≠ aj not all elements of array a are equal ∃ i: 0<i<N: ai > a0 Examples
- 54. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai ≠ aj not all elements of array a are equal ∃ i: 0<i<N: ai > a0 a0 is not the greatest element of array a Examples
- 55. ∀ i: 0≤i<N: ai=0 all elements ai of array a (of length N) are zero ∀ i: 1≤i<N: ai-1 ≤ ai elements of array a are monotonically increasing ∀ i,j: 0≤i<j<N: ai ≤ aj elements of array a are monotonically increasing ∀ i,j: 0≤i,j<N: ai = aj all elements of array a are equal ∀ i: 0<i<N: ai ≤ a0 a0 is the greatest element of array a ∃ i: 0≤i<N: ai ≠ 0 at least one element of array a (of length N) is not zero ∃ i: 1≤i<N: ai-1 > ai elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai > aj elements of array a are not monotonically increasing ∃ i,j: 0≤i<j<N: ai ≠ aj not all elements of array a are equal ∃ i: 0<i<N: ai > a0 a0 is not the greatest element of array a Examples
- 56. ¬(∃ x: U(x): F(x)) ∀ x: U(x): ¬ F(x) ¬(∀ x: U(x): F(x)) ∃ x: U(x): ¬ F(x) De Morgan’s Axiom ¬∃ : De Morgan’s Axiom ¬∀ :
- 57. Example What is the negation of the statement “Always, when he drives then he is sober.”
- 58. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t.
- 59. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t. S(t): He is sober at moment t.
- 60. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t. S(t): He is sober at moment t. ∀t : D(t)->S(t)
- 61. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t. S(t): He is sober at moment t. ∀t : D(t)->S(t) ¬(∀t :: D(t)->S(t))
- 62. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t. S(t): He is sober at moment t. ∀t : D(t)->S(t) ¬(∀t :: D(t)->S(t)) ¬(∀t :: ¬D(t)∨S(t))
- 63. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t. S(t): He is sober at moment t. ∀t : D(t)->S(t) ¬(∀t :: D(t)->S(t)) ¬(∀t :: ¬D(t)∨S(t)) ∃t :: ¬(¬D(t)∨S(t))
- 64. Example What is the negation of the statement “Always, when he drives then he is sober.” D(t): He drives at moment t. S(t): He is sober at moment t. ∀t : D(t)->S(t) ¬(∀t :: D(t)->S(t)) ¬(∀t :: ¬D(t)∨S(t)) ∃t :: ¬(¬D(t)∨S(t)) ∃t :: D(t)∧¬S(t))
- 66. True, False are WFFs. Propositional variables are WFFs. Predicates with variables are WFFs. If A and B are WFFs, then ¬A, A∧B, A∨B, A B, A B are also WFFs. If x is a variable in universe U and A is a WFF, then ∀x:U(x):A(x) and ∃x:U(x):A(x) are also WFFs. Well formed formulas (WFF)
- 67. Bounded variables (∃j: 0<j<N: P(i,j) ∨ Q(i,j)) Free variables If a variable is not bound in a WFF then it is free. An appearance of variable is bound in a WFF if a specific value is assigned to it, or it is quantified. ∃i: 0<i<N: ( ∃i,j: 0<i,j<N: P(i,j) ∨ Q(i,j)
- 68. Bounded variables (∃j: 0<j<N: P(i,j) ∨ Q(i,j)) Free variables If a variable is not bound in a WFF then it is free. An appearance of variable is bound in a WFF if a specific value is assigned to it, or it is quantified. ∃i: 0<i<N: ( ) ∃i,j: 0<i,j<N: P(i,j) ∨ Q(i,j)
- 69. ∃x: U1(x): (∀y: U2(y): Q(x,y)) ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∃x: U1(x): (∀y: U2(y): Q(x,y)) ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∃x: U1(y): (∀y: U2(y): Q(x,y)) Swapping ∀ and ∃ ∀x: U1(x): (∀y: U2(y): Q(x,y)) ∀y: U2(y): (∀x: U1(x): Q(x,y)) ∃x: U1(x): (∃y: U2(y): Q(x,y)) ∃y: U2(y): (∃x: U1(x): Q(x,y)) However
- 70. ∃x: U1(x): (∀y: U2(y): Q(x,y)) ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∃x: U1(x): (∀y: U2(y): Q(x,y)) ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∃x: U1(y): (∀y: U2(y): Q(x,y)) Swapping ∀ and ∃ ∀x: U1(x): (∀y: U2(y): Q(x,y)) ∀y: U2(y): (∀x: U1(x): Q(x,y)) ∃x: U1(x): (∃y: U2(y): Q(x,y)) ∃y: U2(y): (∃x: U1(x): Q(x,y)) However
- 71. Example Every boy loves some girl does not imply that some girl is loved by every boy. U1 - the universe of boys. U2 - the universe of girls. Q(x,y) - boy x loves girl y. ∀y: U2(y): (∃x: U1(x): Q(x,y)) ∃x: U1(y): (∀y: U2(y): Q(x,y))
- 72. ∀x: U(x): (Q(x) ∨ P) (∀x: U(x): Q(x)) ∨ P ∀x: U(x): (Q(x) ∧ P) (∀x: U(x): Q(x)) ∧ P ∃x: U(x): (Q(x) ∨ P) (∃x: U(x): Q(x)) ∨ P ∃x: U(x): (Q(x) ∧ P) (∃x: U(x): Q(x)) ∧ P If P does not contain x as a free variable
- 73. Example Given: an array a[0..N-1] of length N≥3 Find: i, j, k such that: i is the index of the greatest element of a; j is the index of the 2nd greatest element of a; k is the index of the 3rd greatest element of a.
- 74. Example Given: an array a[0..N-1] of length N≥3 Find: i, j, k such that: i is the index of the greatest element of a; j is the index of the 2nd greatest element of a; k is the index of the 3rd greatest element of a. ∃ i,j,k: (0≤i,j,k<N)∧(i≠j)∧(i≠k)∧(j≠k): ( (ai≥aj≥ak) ∧ (∀m: (0≤m<N)∧(m≠i)∧(m≠j)∧(m≠k): ak≥am) )
- 75. Example All elements of an array a of length N are either zero or one
- 76. Example All elements of an array a of length N are either zero or one ∀i: 0≤i<N: ( ai=0 ∨ ai=1 )
- 77. Example All elements of an array a of length N are either zero or one ∀i: 0≤i<N: ( ai=0 ∨ ai=1 ) (∀i: 0≤i<N: ai=0) ∨ (∀i: 0≤i<N: ai=1) (1) (2)
- 78. Example All elements of an array a of length N are either zero or one ∀i: 0≤i<N: ( ai=0 ∨ ai=1 ) (∀i: 0≤i<N: ai=0) ∨ (∀i: 0≤i<N: ai=1) (1) (2) ambiguous
- 79. Example The elements of arrays a and b are the same (both arrays of length N).
- 80. Example The elements of arrays a and b are the same (both arrays of length N). ∀i: 0≤i<N: (ai=bi)(1)
- 81. Example The elements of arrays a and b are the same (both arrays of length N). ∀i: 0≤i<N: (ai=bi) ∀i: 1≤i<N: ((ai=ai-1) ∧ (bi=bi-1)) (1) (2)
- 82. Example The elements of arrays a and b are the same (both arrays of length N). ∀i: 0≤i<N: (ai=bi) ∀i: 1≤i<N: ((ai=ai-1) ∧ (bi=bi-1)) (1) (2) (∀i: 1≤i<N: (ai=ai-1)) ∧ (∀i: 0≤i<N: (ai=bi))(3)
- 83. Example The elements of arrays a and b are the same (both arrays of length N). ∀i: 0≤i<N: (ai=bi) ∀i: 1≤i<N: ((ai=ai-1) ∧ (bi=bi-1)) (1) (2) (∀i: 1≤i<N: (ai=ai-1)) ∧ (∀i: 0≤i<N: (ai=bi))(3) ∀i: 0≤i<N: (∃ j: 0≤j<N: (ai=bi) )(4)
- 84. Example The elements of arrays a and b are the same (both arrays of length N). ∀i: 0≤i<N: (ai=bi) ∀i: 1≤i<N: ((ai=ai-1) ∧ (bi=bi-1)) (1) (2) ambiguous (∀i: 1≤i<N: (ai=ai-1)) ∧ (∀i: 0≤i<N: (ai=bi))(3) ∀i: 0≤i<N: (∃ j: 0≤j<N: (ai=bi) )(4)
- 85. Example
- 86. They drank two cups of tea because they were warm. Example
- 87. They drank two cups of tea because they were warm. They drank two cups of tea because they were cold. Example
- 88. Go to the shop and buy bread. If they have eggs, buy 10. Example
- 89. Go to the shop and buy bread. If they have eggs, buy 10. ... Do you have eggs? Yes. Then give me 10 pieces of bread. Example
- 90. Min x: U(x): T(x) gives the “smallest value of terms T(x) for all objects x from the universe U“
- 91. Min x: U(x): T(x) gives the “smallest value of terms T(x) for all objects x from the universe U“ Min x: False: T(x) is +∞.
- 92. Min x: U(x): T(x) gives the “smallest value of terms T(x) for all objects x from the universe U“ Min x: False: T(x) is +∞. ( m = (Min i: 0≤i<N: a[i]) )
- 93. Min x: U(x): T(x) gives the “smallest value of terms T(x) for all objects x from the universe U“ Min x: False: T(x) is +∞. ( m = (Min i: 0≤i<N: a[i]) ) ((∃ i: 0≤i<N: m=a[i]) ∧ (∀ i: 0≤i<N: m≤a[i]))
- 94. Max x: U(x): T(x) gives the “greatest value of terms T(x) for all objects x from the universe U“
- 95. Max x: U(x): T(x) gives the “greatest value of terms T(x) for all objects x from the universe U“ Max x: False: T(x) is -∞.
- 96. Max x: U(x): T(x) gives the “greatest value of terms T(x) for all objects x from the universe U“ Max x: False: T(x) is -∞. Max x: U(x): T(x) = - (Min x: U(x): - T(x))
- 97. Max x: U(x): T(x) gives the “greatest value of terms T(x) for all objects x from the universe U“ Max x: False: T(x) is -∞. Max x: U(x): T(x) = - (Min x: U(x): - T(x)) Min x: U(x): T(x) = - (Max x: U(x): - T(x))
- 98. Anz x: U(x): F(x) gives the “number of all those objects x from the universe U for which F(x) holds“
- 99. Anz x: U(x): F(x) gives the “number of all those objects x from the universe U for which F(x) holds“ Anz x: False: F(x) is 0.
- 100. Anz x: U(x): F(x) gives the “number of all those objects x from the universe U for which F(x) holds“ Anz x: False: F(x) is 0. ((Anz x: U(x): F(x)) = 0) ∀x: U(x):¬F(x)
- 101. Anz x: U(x): F(x) gives the “number of all those objects x from the universe U for which F(x) holds“ Anz x: False: F(x) is 0. ((Anz x: U(x): F(x)) = 0) ∀x: U(x):¬F(x) ((Anz x: U(x): F(x)) ≥ 0) ∃x: U(x): F(x)
- 102. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. Example
- 103. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 Example
- 104. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 Example
- 105. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 (Anz i: 0≤i<N: ai=x) > 0 Example
- 106. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 (Anz i: 0≤i<N: ai=x) > 0 at least one element ai has the value x (x∈N) Example
- 107. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 (Anz i: 0≤i<N: ai=x) > 0 at least one element ai has the value x (x∈N) (Anz i: 0≤i<N: ai<x) = (Anz i: 0≤i<N: ai>x) Example
- 108. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 (Anz i: 0≤i<N: ai=x) > 0 at least one element ai has the value x (x∈N) (Anz i: 0≤i<N: ai<x) = (Anz i: 0≤i<N: ai>x) x is the median value of a Example
- 109. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 (Anz i: 0≤i<N: ai=x) > 0 at least one element ai has the value x (x∈N) (Anz i: 0≤i<N: ai<x) = (Anz i: 0≤i<N: ai>x) x is the median value of a ∀x:: ( (Anz i: 0≤i<N: ai=x) = (Anz i: 0≤i<N: bi=x) ) Example
- 110. Given arrays a and b. Both arrays are of length N. The elements of both arrays are natural numbers. (Anz i: 0≤i<N: ai=0) = 0 all elements ai of a are different than 0 (Anz i: 0≤i<N: ai=x) > 0 at least one element ai has the value x (x∈N) (Anz i: 0≤i<N: ai<x) = (Anz i: 0≤i<N: ai>x) x is the median value of a ∀x:: ( (Anz i: 0≤i<N: ai=x) = (Anz i: 0≤i<N: bi=x) ) a is a permutation of b Example