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| In [[mathematics]], '''contour sets''' [[Generalization|generalize]] and [[Formal system|formalize]] the everyday notions of
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| *everything superior to something
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| *everything superior or equivalent to something
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| *everything inferior to something
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| *everything inferior or equivalent to something.
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| == Formal definitions ==
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| Given a [[Relation (mathematics)|relation]] on pairs of [[Element (mathematics)|element]]s of [[Set (mathematics)|set]] <math>X</math>
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| :<math>\succcurlyeq~\subseteq~X^2</math> | |
| and an element <math>x</math> of <math>X</math>
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| :<math>x\in X</math> | |
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| The '''upper contour set''' of <math>x</math> is the set of all <math>y</math> that are related to <math>x</math>:
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| :<math>\left\{ y~\backepsilon~y\succcurlyeq x\right\}</math>
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| The '''lower contour set''' of <math>x</math> is the set of all <math>y</math> such that <math>x</math> is related to them: | |
| :<math>\left\{ y~\backepsilon~x\succcurlyeq y\right\}</math>
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| The '''strict upper contour set''' of <math>x</math> is the set of all <math>y</math> that are related to <math>x</math> without <math>x</math> being ''in this way'' related to any of them:
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| :<math>\left\{ y~\backepsilon~(y\succcurlyeq x)\land\lnot(x\succcurlyeq y)\right\}</math>
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| The '''strict lower contour set''' of <math>x</math> is the set of all <math>y</math> such that <math>x</math> is related to them without any of them being ''in this way'' related to <math>x</math>:
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| :<math>\left\{ y~\backepsilon~(x\succcurlyeq y)\land\lnot(y\succcurlyeq x)\right\}</math>
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| The formal expressions of the last two may be simplified if we have defined
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| :<math>\succ~=~\left\{ \left(a,b\right)~\backepsilon~\left(a\succcurlyeq b\right)\land\lnot(b\succcurlyeq a)\right\}</math>
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| so that <math>a</math> is related to <math>b</math> but <math>b</math> is ''not'' related to <math>a</math>, in which case the strict upper contour set of <math>x</math> is
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| :<math>\left\{ y~\backepsilon~y\succ x\right\}</math> | |
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| and the strict lower contour set of <math>x</math> is
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| :<math>\left\{ y~\backepsilon~x\succ y\right\}</math>
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| === Contour sets of a function ===
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| In the case of a [[Function (mathematics)|function]] <math>f()</math> considered in terms of relation <math>\triangleright</math>, reference to the contour sets of the function is implicitly to the contour sets of the implied relation
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| :<math>(a\succcurlyeq b)~\Leftarrow~[f(a)\triangleright f(b)]</math>
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| == Examples ==
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| === Arithmetic ===
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| Consider a [[real number]] <math>x</math>, and the relation [[Inequality (mathematics)|<math>\ge</math>]]. Then
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| * the upper contour set of <math>x</math> would be the set of numbers that were ''greater than or equal'' to <math>x</math>,
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| * the ''strict'' upper contour set of <math>x</math> would be the set of numbers that were ''greater'' than <math>x</math>,
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| * the lower contour set of <math>x</math> would be the set of numbers that were ''less than or equal'' to <math>x</math>, and
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| * the ''strict'' lower contour set of <math>x</math> would be the set of numbers that were ''less'' than <math>x</math>.
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| Consider, more generally, the relation
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| :<math>(a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)]</math>
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| Then
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| * the upper contour set of <math>x</math> would be the set of all <math>y</math> such that <math>f(y)\ge f(x)</math>,
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| * the ''strict'' upper contour set of <math>x</math> would be the set of all <math>y</math> such that <math>f(y)>f(x)</math>,
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| * the lower contour set of <math>x</math> would be the set of all <math>y</math> such that <math>f(x)\ge f(y)</math>, and
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| * the ''strict'' lower contour set of <math>x</math> would be the set of all <math>y</math> such that <math>f(x)>f(y)</math>.
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| It would be ''technically'' possible to define contour sets in terms of the relation
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| :<math>(a\succcurlyeq b)~\Leftarrow~[f(a)\le f(b)]</math>
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| though such definitions would tend to confound ready understanding.
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| In the case of a real-valued function <math>f()</math> (whose arguments might or might not be themselves real numbers), reference to the contour sets of the function is implicitly to the contour sets of the relation
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| :<math>(a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)]</math>
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| Note that the arguments to <math>f()</math> might be [[Tuple|vector]]s, and that the [[Mathematical notation|notation]] used might instead be
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| :<math>[(a_1 ,a_2 ,\ldots)\succcurlyeq(b_1 ,b_2 ,\ldots)]~\Leftarrow~[f(a_1 ,a_2 ,\ldots)\ge f(b_1 ,b_2 ,\ldots)]</math>
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| === Economic ===
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| In [[economics]], the set <math>X</math> could be interpreted as a set of [[Good (economics and accounting)|goods and services]] or of possible [[Outcome (game theory)|outcomes]], the relation <math>\succ</math> as ''strict [[preference]]'', and the relationship <math>\succcurlyeq</math> as ''weak preference''. Then
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| * the upper contour set, or '''better set''',<ref name=":0">{{Cite book|title = Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory|last = Robert P. Gilles|first = |publisher = Springer|year = 1996|isbn = |location = |pages = 35|url = http://books.google.com/books?id=ZyahaTvMB3cC&lpg=PA35&ots=4CelGh9izH&dq=%22better%20set%22%20economics&pg=PA35#v=onepage&q=%22better%20set%22%20}}</ref> of <math>x</math> would be the set of all goods, services, or outcomes that were ''at least as desired'' as <math>x</math>,
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| * the ''strict'' upper contour set of <math>x</math> would be the set of all goods, services, or outcomes that were ''more desired'' than <math>x</math>,
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| * the lower contour set, or '''worse set''',<ref name=":0" /> of <math>x</math> would be the set of all goods, services, or outcomes that were ''no more desired'' than <math>x</math>, and
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| * the ''strict'' lower contour set of <math>x</math> would be the set of all goods, services, or outcomes that were ''less desired'' than <math>x</math>.
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| Such preferences might be captured by a [[utility]] function <math>u()</math>, in which case
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| * the upper contour set of <math>x</math> would be the set of all <math>y</math> such that <math>u(y)\ge u(x)</math>,
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| * the ''strict'' upper contour set of <math>x</math> would be the set of all <math>y</math> such that <math>u(y)>u(x)</math>,
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| * the lower contour set of <math>x</math> would be the set of all <math>y</math> such that <math>u(x)\ge u(y)</math>, and
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| * the ''strict'' lower contour set of <math>x</math> would be the set of all <math>y</math> such that <math>u(x)>u(y)</math>.
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| == Complementarity ==
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| On the assumption that <math>\succcurlyeq</math> is a [[total order]]ing of <math>X</math>, the [[Complement (set theory)|complement]] of the upper contour set is the strict lower contour set.
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| :<math>X^2\backslash\left\{ y~\backepsilon~y\succcurlyeq x\right\}=\left\{ y~\backepsilon~x\succ y\right\}</math> | |
| :<math>X^2\backslash\left\{ y~\backepsilon~x\succ y\right\}=\left\{ y~\backepsilon~y\succcurlyeq x\right\}</math>
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| and the complement of the strict upper contour set is the lower contour set.
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| :<math>X^2\backslash\left\{ y~\backepsilon~y\succ x\right\}=\left\{ y~\backepsilon~x\succcurlyeq y\right\}</math>
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| :<math>X^2\backslash\left\{ y~\backepsilon~x\succcurlyeq y\right\}=\left\{ y~\backepsilon~y\succ x\right\}</math>
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| == See also ==
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| *[[Epigraph (mathematics)|Epigraph]]
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| *[[Hypograph (mathematics)|Hypograph]]
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| ==References==
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| <references /> | |
| == Bibliography ==
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| * [[Andreu Mas-Colell]], Michael D. Whinston, and Jerry R. Green, ''Microeconomic Theory'' ({{LCC|HB172.M6247 1995}}), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)
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| <!-- Yeah, we could use more references, but I'm going to post this article now. -->
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| [[Category:Mathematical relations]]
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| [[Category:Microeconomics]]
| |
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