12.0 KB · ⇊ 7 parts · ⤓ Micron 12.0 KB · ⤓ HTML 95.1 KB

top

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.

• Formal definitions

• Contour sets of a function

• Examples

• Arithmetic

• Economics

• Complementarity

• See also

• References

• Bibliography

Given a relation on pairs of elements of set X {\displaystyle X}

≽ ≽ ⊆ ⊆ X 2 {\displaystyle \succcurlyeq ~\subseteq ~X^{2}}

and an element x {\displaystyle x} of X {\displaystyle X}

x ∈ ∈ X {\displaystyle x\in X}

The upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x {\displaystyle x} :

{ y ∍ ∍ y ≽ ≽ x } {\displaystyle \left\{y~\backepsilon ~y\succcurlyeq x\right\}}

The lower contour set of x {\displaystyle x} is the set of all y {\displaystyle y} such that x {\displaystyle x} is related to them:

{ y ∍ ∍ x ≽ ≽ y } {\displaystyle \left\{y~\backepsilon ~x\succcurlyeq y\right\}}

The strict upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x {\displaystyle x} without x {\displaystyle x} being in this way related to any of them:

{ y ∍ ∍ ( y ≽ ≽ x ) ∧ ∧ ¬ ¬ ( x ≽ ≽ y ) } {\displaystyle \left\{y~\backepsilon ~(y\succcurlyeq x)\land \lnot (x\succcurlyeq y)\right\}}

The strict lower contour set of x {\displaystyle x} is the set of all y {\displaystyle y} such that x {\displaystyle x} is related to them without any of them being in this way related to x {\displaystyle x} :

{ y ∍ ∍ ( x ≽ ≽ y ) ∧ ∧ ¬ ¬ ( y ≽ ≽ x ) } {\displaystyle \left\{y~\backepsilon ~(x\succcurlyeq y)\land \lnot (y\succcurlyeq x)\right\}}

The formal expressions of the last two may be simplified if we have defined

≻ ≻ = { ( a , b ) ∍ ∍ ( a ≽ ≽ b ) ∧ ∧ ¬ ¬ ( b ≽ ≽ a ) } {\displaystyle \succ ~=~\left\{\left(a,b\right)~\backepsilon ~\left(a\succcurlyeq b\right)\land \lnot (b\succcurlyeq a)\right\}}

so that a {\displaystyle a} is related to b {\displaystyle b} but b {\displaystyle b} is not related to a {\displaystyle a} , in which case the strict upper contour set of x {\displaystyle x} is

{ y ∍ ∍ y ≻ ≻ x } {\displaystyle \left\{y~\backepsilon ~y\succ x\right\}}

and the strict lower contour set of x {\displaystyle x} is

{ y ∍ ∍ x ≻ ≻ y } {\displaystyle \left\{y~\backepsilon ~x\succ y\right\}}

In the case of a function f ( ) {\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

( a ≽ ≽ b ) ⇐ ⇐ [ f ( a ) ▹ ▹ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\triangleright f(b)]}

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

• the upper contour set of x {\displaystyle x} would be the set of numbers that were greater than or equal to x {\displaystyle x} ,

• the strict upper contour set of x {\displaystyle x} would be the set of numbers that were greater than x {\displaystyle x} ,

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

• the strict lower contour set of x {\displaystyle x} would be the set of numbers that were less than x {\displaystyle x} .

Consider, more generally, the relation

( a ≽ ≽ b ) ⇐ ⇐ [ f ( a ) ≥ ≥ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}

Then

• the upper contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that f ( y ) ≥ ≥ f ( x ) {\displaystyle f(y)\geq f(x)} ,

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

• the lower contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that f ( x ) ≥ ≥ f ( y ) {\displaystyle f(x)\geq f(y)} , and

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

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

( a ≽ ≽ b ) ⇐ ⇐ [ f ( a ) ≤ ≤ f ( b ) ] {\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 f ( ) {\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

( a ≽ ≽ b ) ⇐ ⇐ [ f ( a ) ≥ ≥ f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}

Note that the arguments to f ( ) {\displaystyle 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 , … … ) ] {\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 )]}

In economics, the set X {\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,cite-ref-0-1-0[1] of x {\displaystyle x} would be the set of all goods, services, or outcomes that were at least as desired as x {\displaystyle x} ,

• the strict upper contour set of x {\displaystyle x} would be the set of all goods, services, or outcomes that were more desired than x {\displaystyle x} ,

• the lower contour set, or worse set,cite-ref-0-1-1[1] of x {\displaystyle x} would be the set of all goods, services, or outcomes that were no more desired than x {\displaystyle x} , and

• the strict lower contour set of x {\displaystyle x} would be the set of all goods, services, or outcomes that were less desired than x {\displaystyle x} .

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

• the upper contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that u ( y ) ≥ ≥ u ( x ) {\displaystyle u(y)\geq u(x)} ,

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

• the lower contour set of x {\displaystyle x} would be the set of all y {\displaystyle y} such that u ( x ) ≥ ≥ u ( y ) {\displaystyle u(x)\geq u(y)} , and

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

On the assumption that ≽ ≽ {\displaystyle \succcurlyeq } is a total ordering of X {\displaystyle X} , the complement of the upper contour set is the strict lower contour set.

X 2 ∖ ∖ { y ∍ ∍ y ≽ ≽ x } = { y ∍ ∍ x ≻ ≻ y } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~y\succcurlyeq x\right\}=\left\{y~\backepsilon ~x\succ y\right\}}

X 2 ∖ ∖ { y ∍ ∍ x ≻ ≻ y } = { y ∍ ∍ y ≽ ≽ x } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~x\succ y\right\}=\left\{y~\backepsilon ~y\succcurlyeq x\right\}}

and the complement of the strict upper contour set is the lower contour set.

X 2 ∖ ∖ { y ∍ ∍ y ≻ ≻ x } = { y ∍ ∍ x ≽ ≽ y } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~y\succ x\right\}=\left\{y~\backepsilon ~x\succcurlyeq y\right\}}

X 2 ∖ ∖ { y ∍ ∍ x ≽ ≽ y } = { y ∍ ∍ y ≻ ≻ x } {\displaystyle X^{2}\backslash \left\{y~\backepsilon ~x\succcurlyeq y\right\}=\left\{y~\backepsilon ~y\succ x\right\}}

• Epigraph

• Hypograph

cite-note-0-11. ↑ citerefrobert-p-gilles1996Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN 9780792342007.

• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (LCC HB172.M6247 1995), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)