# Relation

## Definition

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

$$\rho \subseteq {A}^{2}$$

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

Example:

$$A=\left\{1,3,6\right\}$$

$$AxA=\left\{\left(1,1\right),\left(1,3\right),\left(1,6\right),\left(3,1\right),\left(3,3\right),\left(3,6\right),\left(6,1\right),\left(6,3\right),\left(6,6\right)\right\}$$

$$\rho =\{\left(1,1\right),\left(1,3\right),\left(1,6\right),\left(3,3\right),\left(3,6\right),\left(6,6\right)\}|\rho \subseteq {A}^{2},$$

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

**Properties of Relations**

## Reflexivity

$$(\forall a\in A):a\rho a$$

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

$$(\forall a,b\in A):a\rho b\Rightarrow b\rho a$$

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

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