Lecture Notes, 8 Nov 2010

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§2.3: Functions (cont'd)

§2.4: Sequences

§4.1: Mathematical Induction

1. Function Composition:
   1. Reminder:
      Def:
      Let A,B,C be sets.
      Let g : A → B.
      Let f : B → C.
      Then the composition of f with g
      —denoted "(f o g) : A → C" & read "f of g"—
      =def {(a, c) | (∃b ∈ B)[(g(a) = b) ∧ (f(b) = c)]

      1. i.e., (f o g)(a) = f(g(a))

      2. 
         

      3. E.g.:
         Let f(x) = x+1.
         Let g(y) = 3y.

         Then: (f o g)(z) = f(g(z)) = f(3z) = 3z+1

            but: (g o f)(z) = g(f(z)) = g(z+1) = 3(z+1) = 3z+3.

      4. ∴ Thm: Function composition is not commutative
         • i.e., (f o g) ≠ (g o f)

   2. Thm:
      Let A,B be sets.
      Let f : A → B be a 1– correspondence
      (i.e., total, 1–1 & onto)

      Then (f o f^–1) : B → B = ι_B
         and (f^–1 o f) : A → A = ι_A
         and (f^–1)^–1 = f

   ---

2. Sequences:
   1. Def:
      Let (S = N) ∨ (S = W (= Z⁺)).
      Let T be any non-∅ set.
      Then a is a sequence =def a : S → T.
      • i.e., a sequence is a function from {(0,) 1, 2, …} to any set T
        • usually, T ⊆ R
        • but not always; see Felleisen & Krishnamurthi 2009 for an example of how animation can be viewed as a sequence.

   2. Notation:
      1. a_n for: a(n)
      2. "a_n" is a term of the sequence
      3. {a_n} for: the sequence a : N → T (or the sequence a : W → T)

   3. A sequence is a function.
      A function is a relation.
      A relation is a set.
      ∴ A sequence is a set: {(0, a₀), (1, a₁), …}, where each a_i ∈ T

   4. A sequence can be thought of as an ∞-tuple

   5. Examples:
      1. Example 1:
         0,  1,   4,   9,  16,  25, …
         a₀, a₁, a₂, a₃, a₄,  a₅, …

         i.e., (∀n ∈ N)[a_n = n²]

      2. Example 2 (the Fibonacci sequence — read both links!):
         0,  1,   1,   2,   3,   5,   8,   13, …
         a₀, a₁, a₂, a₃, a₄,  a₅, a₆,   a₇, …

         i.e.:

         a₀ = 0
         a₁ = 1
         (∀n > 1)[a_n = a_n–1 + a_n–2]

   6. General problem:
      1. Given a sequence, to determine its formula
      2. i.e., given I/P & O/P of an algorithm, to determine (the?) an(!) algorithm for it.
      3. Notes:
         1. language learning is a real-life e.g.
         2. in general, ∃ >1 algorithm!
         3. given only a finite, initial sequence of O/P,
            ∃ no way to find the intended algorithm
            • i.e., ∃ no way to predict the future!

         4. See "Computational Learning Theory"

   7. Summations:
      1. A series or summation =def the sum of (some or all) terms in a sequence.

      2. i.e.) a₀ + a₁ + a₂ + …

      3. Notation:
         i = n
         Σa_i for: a₀ + a₁ + … + a_n
         i = 0

   8. For more information on sequences & series ("summations"), link to:
      "Sequences and Summations: Further Information"

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3. Mathematical Induction:
   1. Watch the falling-domino movies at "Recursion & Induction"

   2. Let Falls(x) mean: x falls down.
      Then:

      Falls(domino1);			"base case"
      Falls(domino1)	→	Falls(domino2)
      		∴ Falls(domino2),	by Modus Ponens!
      Falls(domino2)	→	Falls(domino3)
      		∴ Falls(domino3),	by MP
      …
      		∴ Falls(dominok)	"inductive hypothesis"
      Falls(dominok)	→	Falls(dominok+1);	"inductive case"
      		∴ Falls(dominok+1)
      …
      Falls(dominolast–1)	→	Falls(dominolast)
      		∴ Falls(dominolast)
      ∴ ∀n[Falls(dominon)]			"general principle"

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