Contour set

Given a relation on pairs of elements of set $X$

:$\backslash succcurlyeq~\backslash subseteq~X^2$

and an element $x$ of $X$

:$x\backslash in\; X$

The upper contour set of $x$ is the set of all $y$ that are related to $x$:

:$\backslash left\backslash \{\; y~\backslash backepsilon~y\backslash succcurlyeq\; x\backslash right\backslash \}$

The lower contour set of $x$ is the set of all $y$ such that $x$ is related to them:

:$\backslash left\backslash \{\; y~\backslash backepsilon~x\backslash succcurlyeq\; y\backslash right\backslash \}$

The strict upper contour set of $x$ is the set of all $y$ that are related to $x$ without $x$ being in this way related to any of them:

:$\backslash left\backslash \{\; y~\backslash backepsilon~(y\backslash succcurlyeq\; x)\backslash land\backslash lnot(x\backslash succcurlyeq\; y)\backslash right\backslash \}$

The strict lower contour set of $x$ is the set of all $y$ such that $x$ is related to them without any of them being in this way related to $x$:

:$\backslash left\backslash \{\; y~\backslash backepsilon~(x\backslash succcurlyeq\; y)\backslash land\backslash lnot(y\backslash succcurlyeq\; x)\backslash right\backslash \}$

The formal expressions of the last two may be simplified if we have defined

:$\backslash succ~=~\backslash left\backslash \{\; \backslash left(a,b\backslash right)~\backslash backepsilon~\backslash left(a\backslash succcurlyeq\; b\backslash right)\backslash land\backslash lnot(b\backslash succcurlyeq\; a)\backslash right\backslash \}$

so that $a$ is related to $b$ but $b$ is not related to $a$, in which case the strict upper contour set of $x$ is

:$\backslash left\backslash \{\; y~\backslash backepsilon~y\backslash succ\; x\backslash right\backslash \}$

and the strict lower contour set of $x$ is

:$\backslash left\backslash \{\; y~\backslash backepsilon~x\backslash succ\; y\backslash right\backslash \}$

In the case of a function $f()$ considered in terms of relation $\backslash triangleright$, reference to the contour sets of the function is implicitly to the contour sets of the implied relation

:$(a\backslash succcurlyeq\; b)~\backslash Leftarrow~[f(a)\backslash triangleright\; f(b)]$

Consider a real number $x$, and the relation $\backslash ge$. Then

• the upper contour set of $x$ would be the set of numbers that were greater than or equal to $x$,
• the strict upper contour set of $x$ would be the set of numbers that were greater than $x$,
• the lower contour set of $x$ would be the set of numbers that were less than or equal to $x$, and
• the strict lower contour set of $x$ would be the set of numbers that were less than $x$.

Consider, more generally, the relation

:$(a\backslash succcurlyeq\; b)~\backslash Leftarrow~[f(a)\backslash ge\; f(b)]$

Then

• the upper contour set of $x$ would be the set of all $y$ such that $f(y)\backslash ge\; f(x)$,
• the strict upper contour set of $x$ would be the set of all $y$ such that $f(y)>f(x)$,
• the lower contour set of $x$ would be the set of all $y$ such that $f(x)\backslash ge\; f(y)$, and
• the strict lower contour set of $x$ would be the set of all $y$ such that $f(x)>f(y)$.

It would be technically possible to define contour sets in terms of the relation

:$(a\backslash succcurlyeq\; b)~\backslash Leftarrow~[f(a)\backslash le\; f(b)]$

though such definitions would tend to confound ready understanding.

In the case of a real-valued function $f()$ (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

:$(a\backslash succcurlyeq\; b)~\backslash Leftarrow~[f(a)\backslash ge\; f(b)]$

Note that the arguments to $f()$ might be vectors, and that the notation used might instead be

:$[(a\_1\; ,a\_2\; ,\backslash ldots)\backslash succcurlyeq(b\_1\; ,b\_2\; ,\backslash ldots)]~\backslash Leftarrow~[f(a\_1\; ,a\_2\; ,\backslash ldots)\backslash ge\; f(b\_1\; ,b\_2\; ,\backslash ldots)]$

In economics, the set $X$ could be interpreted as a set of goods and services or of possible outcomes, the relation $\backslash succ$ as strict preference, and the relationship $\backslash succcurlyeq$ as weak preference. Then

• the upper contour set, or better set,{{Cite book|title = Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory|last = Robert P. Gilles|publisher = Springer|year = 1996|pages = 35|isbn = 9780792342007|url = https://books.google.com/books?id=ZyahaTvMB3cC&dq=%22better+set%22+&pg=PA35}} of $x$ would be the set of all goods, services, or outcomes that were at least as desired as $x$,
• the strict upper contour set of $x$ would be the set of all goods, services, or outcomes that were more desired than $x$,
• the lower contour set, or worse set, of $x$ would be the set of all goods, services, or outcomes that were no more desired than $x$, and
• the strict lower contour set of $x$ would be the set of all goods, services, or outcomes that were less desired than $x$.

Such preferences might be captured by a utility function $u()$, in which case

• the upper contour set of $x$ would be the set of all $y$ such that $u(y)\backslash ge\; u(x)$,
• the strict upper contour set of $x$ would be the set of all $y$ such that $u(y)>u(x)$,
• the lower contour set of $x$ would be the set of all $y$ such that $u(x)\backslash ge\; u(y)$, and
• the strict lower contour set of $x$ would be the set of all $y$ such that $u(x)>u(y)$.

On the assumption that $\backslash succcurlyeq$ is a total ordering of $X$, the complement of the upper contour set is the strict lower contour set.

:$X^2\backslash backslash\backslash left\backslash \{\; y~\backslash backepsilon~y\backslash succcurlyeq\; x\backslash right\backslash \}=\backslash left\backslash \{\; y~\backslash backepsilon~x\backslash succ\; y\backslash right\backslash \}$

:$X^2\backslash backslash\backslash left\backslash \{\; y~\backslash backepsilon~x\backslash succ\; y\backslash right\backslash \}=\backslash left\backslash \{\; y~\backslash backepsilon~y\backslash succcurlyeq\; x\backslash right\backslash \}$

and the complement of the strict upper contour set is the lower contour set.

:$X^2\backslash backslash\backslash left\backslash \{\; y~\backslash backepsilon~y\backslash succ\; x\backslash right\backslash \}=\backslash left\backslash \{\; y~\backslash backepsilon~x\backslash succcurlyeq\; y\backslash right\backslash \}$

:$X^2\backslash backslash\backslash left\backslash \{\; y~\backslash backepsilon~x\backslash succcurlyeq\; y\backslash right\backslash \}=\backslash left\backslash \{\; y~\backslash backepsilon~y\backslash succ\; x\backslash right\backslash \}$

• Epigraph
• Hypograph

• 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)

Category:Mathematical relations