Partial Orders

A partial order is a relation that satisfies three requirments:

  1. $aRa \forall a \in R$
  2. $aRb$ and $bRa$ implies $a=b$
  3. $aRb$ and $bRc$ implies $aRc$

A nonempty set that satisfies the requirements of a partially order is called partially ordered set.

It might be difficult right now to imagine a set that is partially ordered, but the real numbers under the relation $\leq$ is a partially ordered set. If $aRb$ is an element of this set, like $5 \leq 7$, we write $5 \prec 7.$

two elements of a partially ordered set are said to be comparable if $a \prec b$ or $b \prec a.$
totally ordered
a set is said to be totally ordered if all the elements are comparable

You might want to think of a total ordering with the relation,$<.$

upper bound
an element $c \in A$ is said to be an upper bound for $B \subset A$, if $b \prec c$ for all $b \in B$
least upper bound
$c$ is a least upper bound for $B$ if for all $d$ which are upper bounds of $B$, $c \prec d.$

The mathematical definitions for upper bound and least upper bound correspond to their English translations.

Finally we come to equivalence relations. A relation is said to be an equivalence relation if

  1. If $a \in A$ then $aRa$
  2. $aRb$ implies $a=b$
  3. $aRb$ and $bRc$ implies $aRc.$

Those are the three requirements for an equivalence relation. Equivalence relations differ from a partial order in the second requirement. The transitive property remains the same. You might want to think of the equivalence relations as the = relation which makes sense if you think about the requirements for an equivalence relation.