Editing Well-founded relation (section)

==Reflexivity==

A relation {{mvar|R}} is said to be [[reflexive relation|reflexive]] if {{math|''a'' ''R'' ''a''}} holds for every {{mvar|a}} in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain.  For example, in the natural numbers with their usual order ≤, we have {{nowrap|1 ≥ 1 ≥ 1 ≥ ...}}.  To avoid these trivial descending sequences, when working with a partial order ≤, it is common to apply the definition of well foundedness (perhaps implicitly) to the alternate relation < defined such that {{math|''a'' < ''b''}} if and only if {{math|''a'' ≤ ''b''}} and {{math|''a'' ≠ ''b''}}. More generally, when working with a [[preorder]] ≤, it is common to use the relation < defined such that {{math|''a'' < ''b''}} if and only if {{math|''a'' ≤ ''b''}} and {{math|''b'' ≰  ''a''}}. In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation ≤, which is not.  In some texts, the definition of a well-founded relation is changed from the definition above to include these conventions.