two simple facts about well-founded relations

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

1. 1.

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

2. 2.

The requirement for symmetry is absent, i.e., for each $x,y\in X$, either $xRy$ or $yRx$, 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 TwoSimpleFactsAboutWellfoundedRelations 2013-03-22 18:25:43 2013-03-22 18:25:43 yesitis (13730) yesitis (13730) 10 yesitis (13730) Feature msc 03E20