Relation

Definition

A^{2} = A \times A = \{(a, b) : a \in A, b \in A\}

ρ \subseteq A^{2}

The relation ρ is a set of ordered pairs, a subset of an Cartesian squere AxA.

Example:

A = \{1, 3, 6\}

A x A = \{(1, 1), (1, 3), (1, 6), (3, 1), (3, 3), (3, 6), (6, 1), (6, 3), (6, 6)\}

ρ = {(1, 1), (1, 3), (1, 6), (3, 3), (3, 6), (6, 6)} | ρ \subseteq A^{2},

The relation ρ in this case means ≤(less or equal): (1≤1), (1≤3), (1≤6), (3≤3), (3≤6), (6≤6).

(a \in A) : a ρ a

IN CASE OF ORDERED GRAPH: every vertex of graph hase loop.

Example:ρ is reflexiv, because for every element of A is true: $(1 \leq 1), (3 \leq 3), (6 \leq 6)$

(a, b∈A): aρb⇒bρa

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].

(a, b, c \in A) : a ρ b \land b ρ c \Rightarrow a ρ c

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.

(a, b∈A): aρb⇒bρa

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.

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

Keywords: relations, reflexivity, symmetry, transitivity, anti-symmetry, equivalence, partial order:, dichotomy, order