## Definition 2.3.1: Ordered and Well-Ordered Set

A set

**S**is called**partially ordered**if there exists a relation*r*(usually denoted by the symbol ) between**S**and itself such that the following conditions are satisfied:- reflexive:
*a a*for any element*a*in S - transitive: if
*a b*and*b c*then*a c* - antisymmetric: if
*a b*and*b a*then*a = b*

**S**is called**ordered**if it is partially ordered and every pair of elements*x*and*y*from the set**S**can be compared with each other via the partial ordering relation.
A set **S** is called **well-ordered** if it is an ordered set for which
every non-empty subset contains a smallest element.