Relations and Graphs
The cartesian product of two sets A and B , denoted A X B , is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B .
A relation between two sets is a subset of their cartesian product.
A graph is a pair (S, Γ) where S is a set of nodes and Γ ⊆ S X S .
Properties of relations:
|Reflexive||∀ a (a, a) ∈ R|
|Symmetric||∀ a, b (a, b) ∈ R → (b, a) ∈ R|
|Transitive||∀ a, b, c (a, b) ∈ R ∧ (b, c) ∈ R|
|→ (a, c) ∈ R|
|Antisymmetric||∀ a, b (a, b) ∈ R ∧ (b, a) ∈ R → a = b|
A relation that is reflexive, symmetric, and transitive is an equivalence relation, which corresponds to a partition of the set (a set of disjoint subsets whose union is the set).
A relation that is reflexive, antisymmetric, and transitive is a partial order. Example: ≤ .