Preorder - Wikipedia

CREATED: [2022-02-10 Thu 22:26]
ID: 7449d8d1-75f4-4e22-9a2f-c7518bfe1979
ROAM_REFS: https://en.wikipedia.org/wiki/Preorder
REVIEW_SCORE: 0.0
MTIME: [2024-12-25 Wed 15:54]

Consider a homogeneous relation {\displaystyle \,≤ \,} on some given set {\displaystyle P,} so that by definition, {\displaystyle \,≤ \,} is some subset of {\displaystyle P× P} and the notation {\displaystyle a≤ b} is used in place of {\displaystyle (a,b)∈ \,≤ .} Then {\displaystyle \,≤ \,} is called a preorder or quasiorder if it is reflexive and transitive; that is, if it satisfies:

Reflexivity: {\displaystyle a≤ a} for all {\displaystyle a∈ P,} and Transitivity: if {\displaystyle a≤ b{\text{ and }}b≤ c{\text{ then }}a≤ c} for all {\displaystyle a,b,c∈ P.}

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