In mathematics, contour sets generalize and formalize the everyday notions of
Formal definitions.
Given a relation on pairs of elements of set formula_1
formula_2
and an element formula_3 of formula_1
formula_5
The upper contour set of formula_3 is the set of all formula_7 that are related to formula_3:
formula_9
The lower contour set of formula_3 is the set of all formula_7 such that formula_3 is related to them:
formula_13
The strict upper contour set of formula_3 is the set of all formula_7 that are related to formula_3 without formula_3 being "in this way" related to any of them:
formula_18
The strict lower contour set of formula_3 is the set of all formula_7 such that formula_3 is related to them without any of them being "in this way" related to formula_3:
formula_23
The formal expressions of the last two may be simplified if we have defined
formula_24
so that formula_25 is related to formula_26 but formula_26 is "not" related to formula_25, in which case the strict upper contour set of formula_3 is
formula_30
and the strict lower contour set of formula_3 is
formula_32
Contour sets of a function.
In the case of a function formula_33 considered in terms of relation formula_34, reference to the contour sets of the function is implicitly to the contour sets of the implied relation
formula_35
Examples.
Arithmetic.
Consider a real number formula_3, and the relation formula_37. Then
Consider, more generally, the relation
formula_46
Then
It would be "technically" possible to define contour sets in terms of the relation
formula_59
though such definitions would tend to confound ready understanding.
In the case of a real-valued function formula_33 (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
formula_46
Note that the arguments to formula_33 might be vectors, and that the notation used might instead be
formula_63
Economics.
In economics, the set formula_1 could be interpreted as a set of goods and services or of possible outcomes, the relation formula_65 as "strict preference", and the relationship formula_66 as "weak preference". Then
Such preferences might be captured by a utility function formula_75, in which case
Complementarity.
On the assumption that formula_66 is a total ordering of formula_1, the complement of the upper contour set is the strict lower contour set.
formula_90
formula_91
and the complement of the strict upper contour set is the lower contour set.
formula_92
formula_93