In mathematics, contour sets generalize and formalize the everyday notions of Given a relation on pairs of elements of set
The upper contour set of
: The lower contour set of
is related to them: The strict upper contour set of
being in this way related to any of them: The strict lower contour set of
: The formal expressions of the last two may be simplified if we have defined so that
, in which case the strict upper contour set of
is and the strict lower contour set of
considered in terms of relation
, reference to the contour sets of the function is implicitly to the contour sets of the implied relation Consider a real number
Then Consider, more generally, the relation Then It would be technically possible to define contour sets in terms of the relation though such definitions would tend to confound ready understanding.
In the case of a real-valued function
(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 Note that the arguments to
might be vectors, and that the notation used might instead be In economics, the set
could be interpreted as a set of goods and services or of possible outcomes, the relation
as strict preference, and the relationship
as weak preference.
Then Such preferences might be captured by a utility function
, in which case On the assumption that
, the complement of the upper contour set is the strict lower contour set.
and the complement of the strict upper contour set is the lower contour set.