Discrete Structures

Lecture Notes, 3 Dec 2010

Last Update: 3 December 2010

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§8.2 (pp. 541–542) and §9.1: Graphs, including Digraph Representation of Binary Relations

  1. Graphs:

    1. Defs:

      • A vertex (or: a node) ≈def a geometric point
      • An edge (or: an arc) ≈def a line connecting 2 vertices.

      • Let V be a non-∅ set (of vertices).
        Let E be a set of edges.

        1. G is a graph =def G = (V,E)

          • i.e., a graph is an ordered pair consisting of
            a set of vertices and a set of edges

          • ∴ a graph is just a certain kind of set
            (in fact, a binary relation!)

        2. Let G=(V,E) be a graph.
          Let v1,v2 ∈ V.
          Let e ∈ E.

            v1,v2 are e's endpoints
            and e connects v1 and v2
            e ≈ {v1,v2}

            • Notes:

              1. It would be wrong to say "e={v1,v2}",
                because, ∀v1,v2 ∈ V,
                there can be ≥0 e ∈ E;
                i.e., there can be more than one edge connecting any pair of vertices

              2. More precisely:

                V can be any set whatsoever.
                E is typically a "bag" or "multiset"
                (a set with duplicates)
                of (unordered) pairs of members of V

    2. E.g.)

      1. V = {c | c is a computer on the Internet}
        E = {e | e connects 2 computers on the Internet}

        • i.e.)
            the Internet with all the computers on it is a graph
            (or: it can be modeled as, or modeled by, a graph)

    3. Defs:

        Let G=(V,E) be a graph.

        1. G is a di(rected) graph =def E ⊆ V×V

          • i.e.) E is a (multi-)set of ordered pairs of vertices

        2. e = (v1,v2) ∈ E
          (e isdef a directed edge
          e's initial vertex =def v1
          e's terminal vertex =def v2)

        3. e isdef a loop =def e = (v,v) ∈ E

        • E.g.)

  2. Binary relations can be represented by (or "as") digraphs:

    1. Let R be a binary relation on set A

      • i.e.) R ⊆ A×A

      Let D be a digraph s.t. V = A ∧ E = R.
      Then D represents R in the sense that
      ∃ 1–1 correspondence between D and R

      • i.e.) (∀ proposition PR about R)
                              (∃ proposition PD about D)[PR ↔ PD]

    2. E.g.)

    3. Digraph representations of the properties of binary relations:

      1. R is reflexive:

        • ∀aR(a,a) ↔ (∀v∈V)(∃e)[e is a loop at v]
        • i.e., every vertex has a loop

      2. R is symmetric:

        • (∀a,b)[R(a,b) → R(b,a)] ↔
            (∀v1 ≠ v2)(∀e=(v1,v2))∃e′[e′=(v2,v1)]
        • i.e., every edge has an inverse edge

      3. R is anti-symmetric:

        • (∀a,b)[(R(a,b) ∧ R(b,a)) → a=b] ↔
            (∀v1,v2)[  (∃e1,e2)[e1=(v1,v2) ∧ e2=(v2,v1)]

        • i.e., the only inverse "pairs" are loops

      4. R is transitive:

        • (∀a,b,c)[(R(a,b) ∧ R(b,c)) → R(a,c)] ↔
            (∀v1,v2,v3)[((∃e1=(v1,v2)) ∧ (∃e2=(v2,v3))) →
        • i.e., every sequence of 2 edges has a shortcut

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