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

1. For each $x\in X$ , $x\not R x$ . (See the entry R-minimal element.)
   
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 $R-{minimal}$ element, which can only be $x$ itself. For 2, consider $\{x, y\}$ . It has an $R-{minimal}$ 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.