In mathematics, contour sets generalize and formalize the everyday notions of
everything superior to somethingeverything superior or equivalent to somethingeverything inferior to somethingeverything inferior or equivalent to something.Given a relation on pairs of elements of set X
≽ ⊆ X 2 and an element x of X
x ∈ X The upper contour set of x is the set of all y that are related to x :
{ y ∍ y ≽ x } The lower contour set of x is the set of all y such that x is related to them:
{ y ∍ x ≽ y } 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:
{ y ∍ ( y ≽ x ) ∧ ¬ ( x ≽ y ) } 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 :
{ y ∍ ( x ≽ y ) ∧ ¬ ( y ≽ x ) } The formal expressions of the last two may be simplified if we have defined
≻ = { ( a , b ) ∍ ( a ≽ b ) ∧ ¬ ( b ≽ a ) } 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
{ y ∍ y ≻ x } and the strict lower contour set of x is
{ y ∍ x ≻ y } In the case of a function f ( ) considered in terms of relation ▹ , reference to the contour sets of the function is implicitly to the contour sets of the implied relation
( a ≽ b ) ⇐ [ f ( a ) ▹ f ( b ) ] Consider a real number x , and the relation ≥ . 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 , andthe strict lower contour set of x would be the set of numbers that were less than x .Consider, more generally, the relation
( a ≽ b ) ⇐ [ f ( a ) ≥ f ( b ) ] Then
the upper contour set of x would be the set of all y such that f ( y ) ≥ 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 ) ≥ f ( y ) , andthe 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 ≽ b ) ⇐ [ f ( a ) ≤ 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 ≽ b ) ⇐ [ f ( a ) ≥ f ( b ) ] Note that the arguments to f ( ) might be vectors, and that the notation used might instead be
[ ( a 1 , a 2 , … ) ≽ ( b 1 , b 2 , … ) ] ⇐ [ f ( a 1 , a 2 , … ) ≥ f ( b 1 , b 2 , … ) ] In economics, the set X 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
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 , andthe 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 ) ≥ 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 ) ≥ u ( y ) , andthe strict lower contour set of x would be the set of all y such that u ( x ) > u ( y ) .On the assumption that ≽ is a total ordering of X , the complement of the upper contour set is the strict lower contour set.
X 2 ∖ { y ∍ y ≽ x } = { y ∍ x ≻ y } X 2 ∖ { y ∍ x ≻ y } = { y ∍ y ≽ x } and the complement of the strict upper contour set is the lower contour set.
X 2 ∖ { y ∍ y ≻ x } = { y ∍ x ≽ y } X 2 ∖ { y ∍ x ≽ y } = { y ∍ y ≻ x } 