Misplaced Pages

#546453

13-529: [REDACTED] Look up incomparable in Wiktionary, the free dictionary. Incomparable may refer to: Comparability , in mathematics, with respect to a given relation over a set HMS Incomparable , a proposal for a very large battlecruiser, suggested in 1915 Incomparable (diamond) , one of the largest diamonds ever found Anupama (1966 film) aka "Incomparable" Incomparable (Faith Evans album) ,

26-432: A 2014 album Incomparable (Dead by April album) , a 2011 album See also [ edit ] Comparable (disambiguation) Incomparability property (commutative algebra) Indistinguishability (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Incomparable . If an internal link led you here, you may wish to change

39-463: A symbol indicating comparison, such as < {\displaystyle \,<\,} (or ≤ , {\displaystyle \,\leq \,,} > , {\displaystyle \,>,\,} ≥ , {\displaystyle \geq ,} and many others) is used instead of R , {\displaystyle R,} in which case x < y {\displaystyle x<y}

52-854: Is x {\displaystyle x} is comparable to y {\displaystyle y} if and only if y {\displaystyle y} is comparable to x , {\displaystyle x,} and likewise for incomparability. The comparability graph of a partially ordered set P {\displaystyle P} has as vertices the elements of P {\displaystyle P} and has as edges precisely those pairs { x , y } {\displaystyle \{x,y\}} of elements for which x   = > <   y {\displaystyle x\ {\overset {<}{\underset {>}{=}}}\ y} . When classifying mathematical objects (e.g., topological spaces ), two criteria are said to be comparable when

65-482: Is by definition any subset R {\displaystyle R} of P × P . {\displaystyle P\times P.} Given x , y ∈ P , {\displaystyle x,y\in P,} x R y {\displaystyle xRy} is written if and only if ( x , y ) ∈ R , {\displaystyle (x,y)\in R,} in which case x {\displaystyle x}

78-480: Is defined to be the set of all pairs ( x , y ) ∈ P × P {\displaystyle (x,y)\in P\times P} such that x {\displaystyle x} is comparable to y {\displaystyle y} ; that is, such that at least one of x R y {\displaystyle xRy} and y R x {\displaystyle yRx}

91-645: Is said to be related to y {\displaystyle y} by R . {\displaystyle R.} An element x ∈ P {\displaystyle x\in P} is said to be R {\displaystyle R} -comparable , or comparable ( with respect to R {\displaystyle R} ), to an element y ∈ P {\displaystyle y\in P} if x R y {\displaystyle xRy} or y R x . {\displaystyle yRx.} Often,

104-436: Is true. If the symbol < {\displaystyle \,<\,} is used in place of ≤ {\displaystyle \,\leq \,} then comparability with respect to < {\displaystyle \,<\,} is sometimes denoted by the symbol = > < {\displaystyle {\overset {<}{\underset {>}{=}}}} , and incomparability by

117-460: Is true. A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. Both of the relations comparability and incomparability are symmetric , that

130-584: Is true. Similarly, the incomparability relation on P {\displaystyle P} induced by R {\displaystyle R} is defined to be the set of all pairs ( x , y ) ∈ P × P {\displaystyle (x,y)\in P\times P} such that x {\displaystyle x} is incomparable to y ; {\displaystyle y;} that is, such that neither x R y {\displaystyle xRy} nor y R x {\displaystyle yRx}

143-391: Is written in place of x R y , {\displaystyle xRy,} which is why the term "comparable" is used. Comparability with respect to R {\displaystyle R} induces a canonical binary relation on P {\displaystyle P} ; specifically, the comparability relation induced by R {\displaystyle R}

SECTION 10

#1732773262547

156-434: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Incomparable&oldid=1017551385 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Comparability A binary relation on a set P {\displaystyle P}

169-587: The symbol = > < {\displaystyle {\cancel {\overset {<}{\underset {>}{=}}}}\!} . Thus, for any two elements x {\displaystyle x} and y {\displaystyle y} of a partially ordered set, exactly one of x   = > <   y {\displaystyle x\ {\overset {<}{\underset {>}{=}}}\ y} and x = > < y {\displaystyle x{\cancel {\overset {<}{\underset {>}{=}}}}y}

#546453