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

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

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
 * $$(a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)]$$

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
 * $$(a\succcurlyeq b)~\Leftarrow~[f(a)\le f(b)]$$

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
 * $$(a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)]$$

Note that the arguments to $$f$$ might be vectors, and that the notation used might instead be
 * $$[(a_1 ,a_2 ,\ldots)\succcurlyeq(b_1 ,b_2 ,\ldots)]~\Leftarrow~[f(a_1 ,a_2 ,\ldots)\ge f(b_1 ,b_2 ,\ldots)]$$

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

and the complement of the strict upper contour set is the lower contour set.
 * $$X^2\backslash\left\{ y~\backepsilon~y\succ x\right\}=\left\{ y~\backepsilon~x\succcurlyeq y\right\}$$
 * $$X^2\backslash\left\{ y~\backepsilon~x\succcurlyeq y\right\}=\left\{ y~\backepsilon~y\succ x\right\}$$