Keywords: relations, reflexivity, symmetry, transitivity, anti-symmetry, equivalence, partial order:, dichotomy, order
The relation ρ is a set of ordered pairs, a subset of an Cartesian squere AxA.
The relation ρ in this case means ≤(less or equal): (1≤1), (1≤3), (1≤6), (3≤3), (3≤6), (6≤6).
Properties of Relations
IN CASE OF ORDERED GRAPH: every vertex of graph hase loop.
Example:ρ is reflexiv, because for every element of A is true:
IN CASE OF ORDERED GRAPH: every edge of the graph is two sided.
Example:ρ is not symmetric, because if 3 ≤ 6 [or (3, 6) ∈ ρ], does not follow that 6 ≤ 3 [or (6, 3) ∈ ρ].
IN CASE OF ORDERED GRAPH: if there is a path between two vertices, there is a longer path too between them.
Example:ρ is transitiv, because if 1 ≤ 3 and 3 ≤ 6 means 1 ≤ 6.
IN CASE OF ORDERED GRAPH: there is 0 or 1 edge between any of two different vertecies.
Example:ρ is ant-symmetric, because each number is less than or equal to itself.
Types of Relations
- Equivalence Relations: reflexive, symmetric and transitivev
- Partial order: reflexive, anti-symmetric and transitivevv
- Dichotomy: for every a, b ∈ A, (a, b) ∈ ρ or (b, a) ∈ ρ
IN CASE OF ORDERED GRAPH: there is a path between any two points of the graph.
- Order:, partial order and dichotomy