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2.Properties and Special Relations

08.06.2020

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

  • Reflexive
  • Transitive

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
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