two simple facts about well-founded relations
The following are two simple facts about well-founded relation on :
- 1.
For each , . (See the entry R-minimal element.)
- 2.
The requirement for symmetry is absent, i.e., for each , either or , but not both.
Justifications for these two facts are simple. For 1, consider the subclass . Then has an element, which can only be itself. For 2, consider . It has an element, which is either or , not both.
Fact 1 is provided here for easy reference. Keeping these two facts in mind is helpful when dealing with (proving) basic theorems about the relation.