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Discrete Structures Lecture 26 Predicate Calculus Read Ch 9.1

Discrete Structures Lecture 26 Predicate Calculus Read Ch 9.1. 8.6(c) without using 8.23, part one. ( V i | 0 ≤ i < n+1: b[i]=0)  b[0]=0 V ( V i | 0 < i < n+1: b[i]=0) 0 ≤ i < n+1 = < Remove abbreviation > 0 ≤ i  i < n+1 = < 0 ≤ i  0=i V 0<i >

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Discrete Structures Lecture 26 Predicate Calculus Read Ch 9.1

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  1. Discrete Structures Lecture 26 Predicate Calculus Read Ch 9.1

  2. 8.6(c) without using 8.23, part one (Vi |0 ≤ i < n+1: b[i]=0) b[0]=0V(Vi |0 < i < n+1: b[i]=0) 0 ≤ i < n+1 = < Remove abbreviation > 0 ≤ i  i < n+1 = <0 ≤ i 0=i V 0<i > (0=i V 0<i)  i < n+1 = < (3.46) Distributivity of  over V > (0 = i  i<n+1) V (0 < i  i<n+1) = < Reintroduce abbreviation > (0 = i  i<n+1) V (0 < i<n+1) = < (3.84a) Substitution > (0 = i  0<n+1) V (0 < i<n+1) < Assump 0≤n;0 ≤ n0<n+1;(3.39) Identity of > 0 = i V (0 < i<n+1)

  3. 8.6(c) without using 8.23 (Vi |0 ≤ i < n+1: b[i]=0) b[0]=0V(Vi |0 < i < n+1: b[i]=0) (Vi |0 ≤ i < n+1: b[i]=0) = < Previous Proof > (Vi |0 = iV0 < i < n+1: b[i]=0) = < (8.16) Range Split > (Vi |0 = i: b[i]=0)V(Vi | 0 < i < n+1: b[i]=0) = < (8.14) One Point Rule > b[0]=0 V (Vi |0 < i < n+1: b[i]=0)

  4. A Note about rangesFor example: 2, …, 15 (a) 2 ≤ i ≤ 15 (b) 2 ≤ i < 16 (c) 1 < i ≤ 15 (d) 1 < i < 16 Number of elements in range is equal to upperBound - lowerBound in (b) & (c). Xerox Parc study, said fewest errors with (b).

  5. Predicate Logic An extension of propositional logic that uses variables of types other than B. Propositional Calculus: reasoning about formulas constructed from boolean variables and operators. Predicate Calculus: More expressive class of formulas.

  6. Predicate Calculus Formula Boolean expression in which some Boolean variables may have been replaced by: • Predicates….whose arguments may be of types other than B. • Universal and Existential Quantifiers

  7. Universal Quantification (9.1) (x | Range :P) Read as "for all x such that the Range holds, P holds."  is idempotent, so universal quantification satisfies range split (8.18)….. (8.13)-(8.21) hold as well.

  8. Trading with Universal Quantification (9.2) Axiom, Trading (x|Range:P)  (x|:Range  P) This axiom allows us to prove trading theorems(9.3 a,b,c)and(9.4 a,b,c,d).

  9. Theorem 9.3 (9.3) Theorem, Trading a)(x|R:P)  (x|:¬R V P) b)(x|R:P)  (x|:R  P  R) c) (x|R:P)  (x|:R V P  P)

  10. Theorem 9.4 (9.4) Theorem, Trading a)(x|Q R:P)  (x|Q:R  P) b)(x|Q R:P)  (x|Q:¬R V P) c)(x|Q R:P)  (x|Q:R  P  R) d) (x|Q R:P)  (x|Q:R V P  P)

  11. Distributivity with  (9.5) Axiom, Distributivity of V over  P V (x|R: Q)  (x|R: P V Q) provided ¬occurs ('x','P')

  12. Problem (9.3) Prove theorem (9.6) (9.6) (x | R : P)  P V (x | :¬R) provided ¬occurs ('x','P') (x |R: P) = <(9.3a) Trading > (x | :¬R V P) = <(9.5) V distributes over  > P V (x | :¬R)

  13. More theorems with  (9.7) Distributivity of  over provided ¬occurs ('x','P') ¬(x|:¬R)  ((x|R:PQ)  P  (x|R:Q)) (Antecedent means the range is not everywhere false) Example: ¬(x|:¬(x=0))  ((x|x=0:PQ)  P  (x|x=0:Q)) ¬(x|:x¹0))  ((x|x=0:PQ)  P  (x|x=0:Q)) ¬(x¹0)  ((x|x=0:PQ)  P  (x|x=0:Q)) (x = 0)  ((x|x=0:PQ)  P  (x|x=0:Q)) Likewise: ¬(x|:¬(x>=0))  ((x|x>=0:PQ)  P  (x|x>=0:Q)) ¬(x|: (x<0))  ((x|x>=0:PQ)  P  (x|x>=0:Q)) (x >= 0)  ((x|x>=0:PQ)  P  (x|x>=0:Q))

  14. More theorems with  (9.8) (x | R : true)  true (9.9) (x|R:PQ)  ((x|R:P)  (x|R:Q))

  15. Weakening, Strengthening for  Think back to 3.76a and 3.76b (9.10) Range weakening/strengthening (x | Q V R : P)  (x | Q : P) (9.11) Body weakening/strengthening (x | R : P  Q)  (x | R : P)

  16. Monotonicity of  (9.12) Monotonicity of  (x | R : Q  P)  ((x | R : Q)  (x | R : P))

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