Contour set explained

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

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 ln ot(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 ln ot(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 ln ot(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

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

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,^[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, 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\}

See also

• Epigraph
• Hypograph

References

1. Book: Robert P. Gilles. Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. 1996. 35. 9780792342007.

Bibliography

• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory, p43. (cloth) (paper)