Contour set

Last updated December 10, 2021

In mathematics, contour sets generalize and formalize the everyday notions of

Formal definitions

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

\succcurlyeq ~\subseteq ~X^{2}

and an element $x$ of $X$

x\in X

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

\left\{y~\backepsilon ~y\succcurlyeq x\right\}

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

\left\{y~\backepsilon ~x\succcurlyeq y\right\}

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:

\left\{y~\backepsilon ~(y\succcurlyeq x)\land \lnot (x\succcurlyeq y)\right\}

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 $x$ :

\left\{y~\backepsilon ~(x\succcurlyeq y)\land \lnot (y\succcurlyeq x)\right\}

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

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

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

\left\{y~\backepsilon ~y\succ x\right\}

and the strict lower contour set of $x$ is

\left\{y~\backepsilon ~x\succ y\right\}

Contour sets of a function

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

(a\succcurlyeq b)~\Leftarrow ~[f(a)\triangleright f(b)]

Examples

Arithmetic

Consider a real number $x$ , and the relation $\geq$ . 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\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]

Then

the upper contour set of $x$ would be the set of all $y$ such that $f(y)\geq f(x)$ ,
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 $f(x)\geq f(y)$ , 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\succcurlyeq b)~\Leftarrow ~[f(a)\leq 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\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]

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

[(a_{1},a_{2},\ldots )\succcurlyeq (b_{1},b_{2},\ldots )]~\Leftarrow ~[f(a_{1},a_{2},\ldots )\geq f(b_{1},b_{2},\ldots )]

Economics

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

the upper contour set, or better set,^[1] 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,^[1] 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 $u(y)\geq u(x)$ ,
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 $u(x)\geq u(y)$ , and
the strict lower contour set of $x$ would be the set of all $y$ such that $u(x)>u(y)$ .

Complementarity

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

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

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

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

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

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

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References

1 2 Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN 9780792342007.

Bibliography

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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[:0-1] 1 2 Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN 9780792342007.

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