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:

1. reflexive: a a for any element a in S
2. transitive: if a b and b c then a c
3. antisymmetric: if a b and b a then a = b

A set 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.

Context