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.
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
, 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
≽
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
)
, 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
≽
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
, 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
)
≥
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
)
, and
the 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
}
