two simple facts about well-founded relations

The following are two simple facts about well-founded relation $R$ on $X$ :

1.

For each $x\in X$ , $x\not Rx$ . (See the entry R-minimal element.)
2.

The requirement for symmetry is absent, i.e., for each $x,y\in X$ , either $x R y$ or $y R x$ , but not both.

Justifications for these two facts are simple. For 1, consider the subclass $\{x\}$ . Then $\{x\}$ has an $R-\textrm{minimal}$ element, which can only be $x$ itself. For 2, consider $\{x,y\}$ . It has an $R-\textrm{minimal}$ element, which is either $x$ or $y$ , 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.

Title	two simple facts about well-founded relations
Canonical name	TwoSimpleFactsAboutWellfoundedRelations
Date of creation	2013-03-22 18:25:43
Last modified on	2013-03-22 18:25:43
Owner	yesitis (13730)
Last modified by	yesitis (13730)
Numerical id	10
Author	yesitis (13730)
Entry type	Feature
Classification	msc 03E20