Lecture Notes, 1 Nov 2010

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§2.2: Set Operations (cont'd)

§2.3: Functions

1. Reminder: Last time, I introduced some notation that cannot easily be shown in HTML; I'll do my best.

   1. Notation:

      i = n
      ∪A_i =def A₁ ∪ A₂ ∪ … ∪ A_n
      i = 1

      = (alternative notation)

        n
      ∪A_i
      i = 1

      = (alternative notation)

        n
      ∪A_i
        1

      = (alternative notation) ∪_{i ∈ {1,…, n}}A_i = (alternative notation) ∪_iA_i

      • (which should remind you of ∃iA(i))

   2. E.g.:
      Suppose (∀i ∈ {1,…, n})[A_i = {i}]
      ∴ A₁ = {1}, A₂ = {2}, …

      1. Then:
          4
         ∪A_i = {1} ∪ {2} ∪ {3} ∪ {4} = {1,2,3,4}
         i=1

         and

          n
         ∪A_i = {1,2,3,…, n}
         i=1

         and

          ∞
         ∪A_i = ∪_{i ∈ W}A_i = {1} ∪ … ∪ {i} ∪ … = W
         i=1

   3. Similarly:
      i=n
      ∩A_i = A₁ ∩ … ∩ A_n
      i=1
      • E.g.:
        i=4
        ∩A_i = {1} ∩ {2} ∩ {3} ∩ {4} = ∅
        i=1

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2. Be sure to read "Using Sets to Define the Natural Numbers",

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3. Functions:
   1. Note about the textbook:
      • The book defines "function" in terms of an "assignment";
        but "assignment" is never defined!

      • Nor does the book define "function" in terms of "relations";
        but I will :-)

   2. Relations:
      1. Informally, a binary relation between 2 objects is like a property that belongs to the ordered pair of the objects

         E.g.:

         1. x < y (e.g., 1 < 2)
            • "<" is a 2-place relation between x & y
              (or: it's a property of (x, y) )

         2. x is a sibling of y (Kim is Pat's sibling)
            • "being a sibling of" is a 2-place relation between x & y

         3. x is a course that is taken by student y (191 is a course that you take)

         4. x is between y and z (2 is between 1 and 3)

      2. (Formal, Set-Theoretic) Definition:
         Let A,B be sets.
         Then R is a binary relation on A × B
         or: R is a binary relation from A to B
         or: R is a binary relation between A and B

         =def R ⊆ A × B

         • i.e., R ⊆ {(a, b) | (a ∈ A) ∧ (b ∈ B)}
         

   3. Functions:
      1. Basic idea: A function is a relation s.t. the same input (I/P) always has the same output (O/P)
         • So, the graphs of curves like a straight line with slope = 1, or a parabola open at the top, or a sine wave are functions.
         • But a circle is not a function.
         • And √ is not a function (each input has two outputs).

           

      2. Def:
         Let A,B be sets.
         Then f is a function from A to B
         =def
         1. f is a binary relation from A to B,
            and
         2. (∀a∈A)(∀b∈B)(∀b′∈B)[( (a,b) ∈ f  ∧  (a,b′) ∈ f ) → b=b′]

      3. i.e., no two distinct members of f have the same first element (but different 2nd elements)

      4. I.e., if "2" members of f have the same first element,
         then they have the same second element.
         • i.e., it only seems as if they are 2 members.
         • i.e., "they" are one, not two

      5. i.e., same I/P → same O/P
         

   4. Notation & Terminology:
      1. "f : A → B" for: f is a function from A to B
         • Note: The "→" is NOT a material-conditional arrow!!!!!

      2. "f(a) = b"
         or                      for: (a, b) ∈ f
         "f : a |→ b"

      3. In the above, "a" and "b" are associated with different jargon in different academic disciplines:

         discipline	a	b
         social sciences	independent variable	dependent variable
         		b is a function of a
         		b depends on a
         mathematics	pre-image of b	image of a
         computer science	input	output

      4. 
         

      5. f "maps" A "to" B

         f is a "transformation of" A "into" B

   5. Def:
      Let A,B be sets.
      Let a,c ∈ A; b,d ∈ B.
      Let f, g : A → B
      Then f = g  =def  {(a, b) | (a, b) ∈ f} = {(c, d) | (c, d) ∈ g}
      1. i.e., f = g  ↔  ∀a∀b[(a, b) ∈ f  ↔  (a, b) ∈ g]

      2. i.e., f = g  ↔  ∀a∀b[f(a) = b  ↔  g(a) = b]

      3. i.e, "2" functions are the same iff they are the same set

   6. The So-Called "Function Machine"
      1. Consider a machine f that:
         • takes input a,
         • you turn a crank,
         • it grinds away at the input,
         • and finally it outputs b (i.e., f(a)):

         

      2. Despite what you may have been told elsewhere (e.g., in high school), this is NOT what a function is!

      3. A function is merely the set of input-output (I/O) pairs.

      4. What the machine really is is a computer!
         • (And the "gears" are an algorithm that computes the function.)
         • But: not all functions can be computed by algorithms!
         • I.e., there are functions for which there are no such "function machines"

      5. For more info, take CSE 396
      6. or see "What Is Computation?"

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