### Contents/Index

1. Intro

@2. Properties and Special Relations

3. Functions as relations

Given a relation $R$ we have that

- $R$ is reflexive if for all $a$ we have that $a\ R\ a$. That is $a$ is related to itself. This is typical for the equal relation: we have that $a = a$. However for the relation "is less then" this does not hold.
- $R$ is symmetric if we have that
$$
a\ R\ b \Rightarrow b\ R\ a
$$
That is if $a$ is related to $b$, then $b$ is related to $a$. This is again typical for the equal relation. However for the relation "is less than" this does not hold.
- $R$ is irreflexive, or anti-reflexive, if for no $a$ we have that $a\ R\ a$. A typical example is the relation "less than". No $a$ is less than itself.
- $R$ is anti-symmetrical if
$$
a\ R\ b \land b\ R\ a \Rightarrow a = b
$$
A typical example is the "less than or equal to". If we have that $a \leq b$ and $b \leq a$, then we have that $a = b$.
- $R$ is asymmetrical if and only if $R$ is both irreflexive and anti-symmetrical.
- $R$ is transitive if we have that
$$
a\ R\ b \land b\ R\ c \Rightarrow a\ R\ c
$$
This holds true for both "less than", "less then or equal to" and "equal to". If $a \lt b$ and $b \lt c$, then sure $a \lt c$.
- $R$ is a connex relation (or has the property of connexity) if for all $a,b$ we have
$$
a\ R\ b \lor b\ R\ a
$$
- $R$ is a semiconnex relation if for all $a,b$ we have
$$
a \neq b \Rightarrow a\ R\ b \lor b\ R\ a
$$

Now we can define special kind of relations.

### Equivalence Relation

An equivalence relation is a relation that is:

- Reflexive
- Symmetric
- Transitive

### Partial Order

A partial order is a relation that is

- Reflexive
- Antisymmetric
- Transitive

A set with a partial order is called a **partially ordered set** or a **poset**.

### Preorder

A preorder relation is a relation that is

### Total Preorder

A total preorder is a relation that is

- Transitive
- Connexitive
- Reflexive

A set that is equipped with a preorder is called a **preordered set**.

### Total Order

A partial order is a relation that is

- Antisymmetric
- Transitive
- Connexitive