# Relation

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

## Definition

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

Example:

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

**Reflexivity**

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

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

**Symmetry**

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].

**Transitivity**

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.

**Anti-symmetry**

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