Contour set

Contour sets of a function

In the case of a function ${\displaystyle f()}$ considered in terms of relation ${\displaystyle \triangleright }$, reference to the contour sets of the function is implicitly to the contour sets of the implied relation

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\triangleright f(b)]}$

Arithmetic

Consider a real number ${\displaystyle x}$, and the relation ${\displaystyle \geq }$. Then

• the upper contour set of ${\displaystyle x}$ would be the set of numbers that were greater than or equal to ${\displaystyle x}$,
• the strict upper contour set of ${\displaystyle x}$ would be the set of numbers that were greater than ${\displaystyle x}$,
• the lower contour set of ${\displaystyle x}$ would be the set of numbers that were less than or equal to ${\displaystyle x}$, and
• the strict lower contour set of ${\displaystyle x}$ would be the set of numbers that were less than ${\displaystyle x}$.

Consider, more generally, the relation

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}$

Then

• the upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(y)\geq f(x)}$,
• the strict upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(y)>f(x)}$,
• the lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(x)\geq f(y)}$, and
• the strict lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(x)>f(y)}$.

It would be technically possible to define contour sets in terms of the relation

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\leq f(b)]}$

though such definitions would tend to confound ready understanding.

In the case of a real-valued function ${\displaystyle f()}$ (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

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}$

Note that the arguments to ${\displaystyle f()}$ might be vectors, and that the notation used might instead be

${\displaystyle [(a_{1},a_{2},\ldots )\succcurlyeq (b_{1},b_{2},\ldots )]~\Leftarrow ~[f(a_{1},a_{2},\ldots )\geq f(b_{1},b_{2},\ldots )]}$

Economics

In economics, the set ${\displaystyle X}$ could be interpreted as a set of goods and services or of possible outcomes, the relation ${\displaystyle \succ }$ as strict preference, and the relationship ${\displaystyle \succcurlyeq }$ as weak preference. Then

• the upper contour set, or better set, of ${\displaystyle x}$ would be the set of all goods, services, or outcomes that were at least as desired as ${\displaystyle x}$,
• the strict upper contour set of ${\displaystyle x}$ would be the set of all goods, services, or outcomes that were more desired than ${\displaystyle x}$,
• the lower contour set, or worse set, of ${\displaystyle x}$ would be the set of all goods, services, or outcomes that were no more desired than ${\displaystyle x}$, and
• the strict lower contour set of ${\displaystyle x}$ would be the set of all goods, services, or outcomes that were less desired than ${\displaystyle x}$.

Such preferences might be captured by a utility function ${\displaystyle u()}$, in which case

• the upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(y)\geq u(x)}$,
• the strict upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(y)>u(x)}$,
• the lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(x)\geq u(y)}$, and
• the strict lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(x)>u(y)}$.