Lecture Notes, 29 Nov 2010

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§§7.1–7.2: Recurrence Relations (cont'd)

1. Answer to the example from last lecture:
   1. 
      Consider this sequence:
      a₀ = C₀ = 0
      a₁ = C₁ = 1

      a_n = 3a_n–1 – 2a_n–2, ∀ n ≥ 2

   2. Execution of our algorithm:
      • I/P: a_n = 3a_n–1 – 2a_n–2
      1. Char Eqn: r² – 3r + 2 = 0

      2. Solve for r:
         • r₁ = (3 + √(9 – 4*2))/2 = (3+√1)/2 = 2

         • r₂ = (3 – √(9 – 4*2))/2 = (3–√1)/2 = 1

      3. Find α_i:
         • a₀ = α₁*2⁰ + α₂*1⁰ = α₁ + α₂ = 0

           ∴ α₁ = –α₂

         • a₁ = α₁*2¹ + α₂*1¹ = 2α₁ + α₂ = 1

           ∴ 2(–α₂) + α₂ = 1

           ∴ –α₂ = 1

           ∴ α₂ = –1

           ∴ α₁ = 1

      • O/P:
        ∴ a_n = α₁r₁ⁿ + α₂r₂ⁿ
                 = 1*2ⁿ + (–1)*1ⁿ
                 = 2ⁿ – 1

   3. You should try this for the other initial conditions shown in the table in the previous lecture to make sure that you get the same answers that I did.

§8.1 and §8.5: Relations

1. In language, including languages from math & CS (such as programming languages), there are:
   1. noun phrases:
      • individual terms & descriptions, including proper names
        • e.g.) 'Fido', 'Prof. Rapaport', '2'

      • general terms and descriptions
        • e.g.) 'dog', 'professor', 'real number'

   2. verb phrases:
      • e.g.) 'sees', 'runs', 'divides', '<'

   3. adjective phrases & prepositional phrases:
      • e.g.) 'is red', 'is even'
   1. We can model these mathematically as follows:
      1. objects in the domain: model individual NPs
      2. sets of objects: model general NPs, intransitive VPs, adjective phrases, properties
      3. n-ary relations (Cartesian products of sets): model transitive VPs, relational properties, prepositions

   ---

2. 1. Def:
      Let A,B be sets.
      Then R is a binary relation on A and B
      =def
                R ⊆ A × B

   2. Notation:
      Let a ∈ A, b ∈ B.

      set-theoretical notation:	(a,b) ∈ R
      FOL notation:	"R(a,b)" for: (a,b) ∈ R "¬R(a,b)" for: (a,b) ∉ R
      mathematical notation:	"aRb" for (a,b) ∈ R

   3. E.g.) Let
      C = {c | c is a US city}
      S = {s | s is a US state}

      Then:

      R = {(c,s) ∈ C×S | c is a city in state s}
      is a binary relation.

      e.g.) (Buffalo, NY) ∈ R

      i.e.) R(Buffalo, NY)

      We could also have a 3-place ("ternary") relation Z ⊆ C×S×ZIP s.t. (Buffalo, NY, 14260) ∈ Z
      or: Z(Buffalo, NY, 14260)

   4. E.g.)
      <  =def  {(x,y) ∈ N×N | y – x ∈ W}

      or, recursively (where S(y) = y+1):

      Base Case:	(0,1) ∈ <
      Recursive Case #1:	(x,y) ∈ <  →  (x,S(y)) ∈ <
      Recursive Case #2:	(x,y) ∈ <  →  (S(x),S(y)) ∈ <

      e.g.)

      Show (2,4) ∈ <:
      Show (2,3) ∈ < (and then use Rec Case #1):
      Show (1,2) ∈ < (and then use Rec Case #2):
      Show (0,1) ∈ <
      (and then use Rec Case #2):
      Trivial! That's the Base Case!

   5. E.g.) The Semantic Web:
      1. Let #Flipper be the Uniform Resource Identifier (URI)
         for Flipper the Dolphin.

         Let #isa be the URI for {(x,y) ∈ U×U | x is a y},
         where U = universe (i.e., domain).

         Let #Dolphin be the URI for the category of dolphins.

      2. Then "#isa(#Flipper, #Dolphin)" represents
         "Flipper is a dolphin".

      3. In addition to making "Flipper" a link,
         could also have semantic information about Flipper,
         to help search engines find Flipper when searching for "dolphin".

      4. The computer languages for expressing this are based on FOL.
         • RDF
         • OWL

      5. For more info, see the 2 articles linked online at "Relations"

   ---

3. Relations have properties:
   1. Def:
      Let A be a set.
      Let R ⊆ A × A be a relation.

      Then:

      1. R is reflexive =def (∀a ∈ A)R(a,a)
      2. R is symmetric =def (∀a,b ∈ A)[R(a,b) → R(b,a)]
      3. R is non-symmetric =def ¬(R is symmetric)
         ≡ ¬(∀a,b ∈ A)[R(a,b) → R(b,a)]
         ≡ (∃a,b ∈ A)[R(a,b) ∧ ¬R(b,a)]

      4. R is anti-symmetric =def
         (∀a,b ∈ A)[ ( R(a,b) ∧ R(b,a) ) → a=b]
      5. R is transitive =def
         (∀a,b,c ∈ A)[ ( R(a,b) ∧ R(b,c) ) → R(a,c)]
      6. R is an equivalence relation =def
         R is reflexive ∧ R is symmetric ∧ R is transitive
      

   2. E.g.)

      rel'n	ref?	sym?	anti-sym?	trans?	equiv?
      ≤	√	×	√	√	×
      ≥	√	×	√	√	×
      <	×	×	√ vacuously!	√	×
      >	×	×	√ vacuously!	√	×
      =	√	√	√	√	√

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