2.Properties and Special Relations

08.06.2020

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.

An equivalence relation is a relation that is:

A partial order is a relation that is

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

A preorder relation is a relation that is

A total preorder is a relation that is

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

A partial order is a relation that is