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单词 TwoSimpleFactsAboutWellfoundedRelations
释义

two simple facts about well-founded relations


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

  1. 1.

    For each xX, xx. (See the entry R-minimal element.)

  2. 2.

    The requirement for symmetry is absent, i.e., for each x,yX, 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 relationMathworldPlanetmath.

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