Lecture Notes, 27 Oct 2010

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§§2.1–2.2: Sets & Set Operations

1. Cardinality:
   1. Def:
      Let S = {e₁, …, e_n}.
      • I.e., S is a finite set!

      Then the cardinality of S =def n

      • I.e., the cardinality of a set is the number of elements that it has.

   2. Notation: |S| for: the cardinality of S

   3. E.g.:
      1. Let Alpha = {A,B,…,Z}.
         Then |Alpha| = 26.

      2. |∅| = 0.
      3. |{0,1,2}| = 3.
      4. |℘({0,1,2})| = 8
         • Note: 8 = 2³. Coincidence? I don't think so!
         • In general, |℘(S)| = 2^|S|

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2. Cartesian Product:
   1. Def:
      Let A,B be sets.
      Then the Cartesian product of A,B =def {(a, b) | (a ∈ A) ∧ (b ∈ B)}

   2. Note: (a, b) is an "ordered pair" with a first element & a second element
      • So, (a, b) ≠ (b, a)

   3. Notation: A × B for: the Cartesian product of A,B

   4. E.g.:
      1. Let S = {a, b}
         and T = {0,1,2}.

         Then S × T = {(s, t) | (s ∈ S) ∧ (t ∈ T)}
                           = {(a,0), (a,1), (a,2), (b,0), (b,1), (b,2)}

         • Note: |S × T| = 6 = |S| * |T| (!)

      2. T × S = {(t, s) | (s ∈ S) ∧ (t ∈ T)}
                   = {(0,a), (1,a), (2,a), (0,b), (1,b), (2,b)}
                   ≠ S × T
         • But: |T × S| = |S × T|

      3. S × S = {(a,a), (a,b), (b,a), (b,b)}

   5. Def:
      Let A₁, …, A_n be sets.
      Then A₁ × … × A_n =def {(a₁, …, a_n) | ∀i[a_i ∈ A_i]}
      • Note the quantification over the subscript!

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3. Ordered n-Tuples:
   1. Def: (Kuratowski's Def of Ordered Pair)
      The ordered pair (a, b) =def { {a}, {a, b} }

   2. Axiom governing this definition:
      (∀a,b,c,d)[(a, b) = (c, d) ↔ [(a = c) ∧ (b = d)]]
      • Note the abbreviation of ∀a∀b∀c∀d as (∀a,b,c,d)

   3. Def:
      • The ordered triple (a,b,c) =def the ordered pair (a, (b, c))
        = { {a}, (b, c) }
        = { {a}, { {b}, {b, c} } }
      • The ordered quadruple (a,b,c,d) =def the ordered pair (a, (b, (c, d)))
      • The ordered quintuple (a,b,c,d,e) =def the ordered pair (a, (b, (c, (d, e))))
      • …
      • The ordered n-tuple or sequence (a₁, … a_n)
        =def the ordered pair (a₁, (a₂, (a₃, … (a_n–1, a_n)…)))
      1. E.g.: Ordered-Alpha = (A,B,C,…,Z) is an ordered 26-tuple.
         • Ordered-Alpha ≠ Alpha = {A,B,C,…,Z} = {Z,Y,X,…A}, etc.

      2. Let POTUS₂₀₁₀ = {GW, JA, TJ, …, GWB, BO}.
         Then |POTUS₂₀₁₀| = 43.
         But BO is the 44th President! How can that be?
         • Ordered-POTUS₂₀₁₀ = (GW, JA, TJ, …, CAA, GC, BH, GC, WMcK, …, GWB, BO),
           where GW was the 1st President,
           …,
           Chester Alan Arthur was the 21st,
           Grover Cleveland (from Buffalo) was the 22nd,
           Benjamin Harrison was the 23rd,
           Grover Cleveland was re-elected as the 24th(!)
           William McKinley (assassinated in Buffalo!) was the 25th,
           …
           and Barack Obama is the 44th.
           • Ordered n-tuples can have duplicate "members".

   4. For more about this, see "Ordered pair" and "Tuple" at Wikipedia

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4. Set Operations:
   1. Defs:
      Let A,B be sets.
      Then:
      1. A ∪ B (the union of A,B) =def {x | (x ∈ A) ∨ (x ∈ B)}

      2. A ∩ B (the intersection of A,B) =def {x | (x ∈ A) ∧ (x ∈ B)}

      3. A and B are disjoint =def A ∩ B = ∅

      4. A – B (the set difference of A,B) =def {x | (x ∈ A) ∧ (x ∉ B)}

      5. Let U be the universal set
         (i.e., the universe, or domain of discourse)

         Then ‾A (the complement of A)

         =def U – A
         = (see note (b), below) {x | (x ∈ U) ∧ (x ∉ A)}
         ≈ (see note (c), below) {x | x ∉ A}
         1. HTML limitations make it impossible, as far as I know,
            to draw the set-complement symbol, i.e., the overline, over the name of the set.

            It should really look something like this:

            ‾
            A

         2. The equality above is a theorem requiring proof.
         3. The wavy, "sort of" equality above is true assuming that the domain is the universe;
            i.e., that everything is, by default, a member of U.

         4. Thm:
            A – B = A ∩ (‾B)

   2. Note:
      ∪ corresponds to ∨
      ∩ corresponds to ∧
      ‾ corresponds to ¬
      Question: What set operation would correspond to →?
      Hint: (p → q) ≡ (¬p ∨ q)

   3. Here's an example I didn't get to cover in lecture:
      Let Alpha = {a,b,…z}.
      Let VOWELS = {a,e,i,o,u,y}.
      Then:
      CONSONANTS = Alpha – V
      VOWELS ∪ CONSONANTS = Alpha.
      VOWELS ∩ CONSONANTS = ∅

   4. On Venn diagrams, see Kosara 2009

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