Contour Set

In mathematics, contour sets generalize and formalize the everyday notions of
*everything superior to something
*everything superior or equivalent to something
*everything inferior to something
*everything inferior or equivalent to something.

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\$

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

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\$

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\$

The formal expressions of the last two may be simplified if we have defined
:$\succ~=~\left\$
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\$

and the strict lower contour set of $x$ is
:$\left\$

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~ (a)\triangleright f(b)/math>

Examples

Arithmetic

Consider a real number $x$, and the relation $\ge$. 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\succcurlyeq b)~\Leftarrow~ (a)\ge f(b)/math>
Then
* the upper contour set of $x$ would be the set of all $y$ such that $f(y)\ge f(x)$,
* 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 $f(x)\ge f(y)$, 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\succcurlyeq b)~\Leftarrow~ (a)\le f(b)/math>
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~ (a)\ge f(b)/math>
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~ (a_1 ,a_2 ,\ldots)\ge f(b_1 ,b_2 ,\ldots)/math>

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, 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 $u(y)\ge u(x)$,
* 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 $u(x)\ge u(y)$, 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.
:$X^2\backslash\left\=\left\$
:$X^2\backslash\left\=\left\$

and the complement of the strict upper contour set is the lower contour set.
:$X^2\backslash\left\=\left\$
:$X^2\backslash\left\=\left\$

See also

* Epigraph
* Hypograph

References

Bibliography

* Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, ''Microeconomic Theory'' (), p43. (cloth) {{isbn, 0-19-510268-1 (paper)

Mathematical relations