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 &isin A and b &isin B .
A relation between two sets is a subset of their cartesian product.
A graph is a pair (S, &Gamma) where S is a set of nodes and &Gamma &sube S X S .
Properties of relations:
| Property: | Definition: |
| Reflexive | &forall a (a, a) &isin R |
| Symmetric | &forall a, b (a, b) &isin R &rarr (b, a) &isin R |
| Transitive | &forall a, b, c (a, b) &isin R &and (b, c) &isin R |
| &rarr (a, c) &isin R | |
| Antisymmetric | &forall a, b (a, b) &isin R &and (b, a) &isin R &rarr 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: &le .