Lecture Notes, 3 Dec 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.
        Then:
        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.
           Then:
           v₁,v₂ are e's endpoints
           and e connects v₁ and v₂
           =def
           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

                • For more info on multisets, see:
                  • "A Set-Theoretical Definition of Graphs"

                  • Rosen, p. 132,
                    following problem #58.

   2. E.g.)

      1. 
         

      2. 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.
      Then:
      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 is_def a directed edge
         ∧
         e's initial vertex =def v₁
         ∧
         e's terminal vertex =def v₂)

      3. e is_def 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 P_R about R)
                              (∃ proposition P_D about D)[P_R ↔ P_D]

   2. E.g.)
      • Let R = {(1,1), (1,2), (1,3), (2,1), (2,2), (3,3)}
        be a relation on {1,2,3,4}.
        
      • Represent each element of {1,2,3,4} as a vertex.
        
      • Represent each ordered pair of R as a directed edge.
        
      • Then the following graph is a representation of R:
        

      • So is:
        

      • So is:
        

   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)]
                                 →
                       v₁=v₂]

         • 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))) →
                                                                   (∃e₃=(v₁,v₃))]
         • i.e., every sequence of 2 edges has a shortcut
         • 
           

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