Contour set

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 \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\} == See also ==