Chu Thanh Duc – MDE 10
Chapter 1. Sets and Mappings
1.1.Elements of logic
-
A Theorem: is simply a statement deduced from other statements and should be a
concept familiar from courses in mathematics.
-
Theorems provice a compact and precise format for presenting the assumptions and
important conclusions of somtimes lengthy arguments, and so help to identify
immediately the scope and limitations of the result presented.
1.1.1
-
Necessity and Sufficiency
Necessity: A is necessary for B  A must hold or be true in order for B hold or be
true  B is true require that A must also be true  “A if B” or “A is implyed by B”
(A  B).
If A is not true, B must be not true. But that doesn’t mean that if B is not true, A must be
not true.
A true
B true
B not true
A not true
 A is not true  B must be not true 
B is not true is necessary for A is not true.
~ A ~ B .
-
Sufficiency: “A is sufficient for B” means that A holds, B must hold. We can say “A
is true only if B is true”; or A implies B (A  B).
B true
A true
A not true
B not true
B is not true  A must be not true.  B is not true is sufficient for A is not true
~B  ~A.
-
Both Necessity and Sufficiency: A  B
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A
1.1.2
B
Theorems and Proofs
“A  B” : A is true  B must be true
-
Here, A is called “premise” and “B is call “conclusion”
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Constructive proof / Direct proof: Assume that A is true, deduce various
consequences of that, and use them to show that B must hold.
-
Contrapositive proof: Assume that B does not hold, then show that A cannot hold.
1.2.Elements of Set Theory
1.2.1 Notation and Basic Concepts
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A Set: is any collection of elements. Elements may be numbers or vectors
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A Subset: Set S is a subset of set T if every element of set S is also an element of set
T. Notation: S  T.
-
Empty set: S is an empty set if it contains no elements at all. Notaton: S = 0
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Complement Set: The complement of set S in an Universal set U is the set of all
elements in U which are not in S. Notation: Complemet set of S: cS.
-
Union: Notation S  T  {x | x  S _ or _ x  T } . In general: U iI Si (I is Index set).
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Intersection: Notation S  T  {x | x  S _ and _ x  T } . Or  iI Si (I is Index set).
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Index Set: The set of interger number starting with 1. S = (1, 2, 3, 4...n). Notation:
I  {1,2,3...}
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The Product of two sets: S  T  {( s, t ) | s  S , t  T }
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n – Space: is the Set Product : R n  R  R  ...  R  {( x1, x2,..., xn) | xi  R} i =1,2...n.
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Non – Negative Orthant: Rn  {( x1, x2,..., xn) | xi  0}  R n
1.2.2 Convex Set:
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The property of convexity is most often assumed  analysis is mathematically tractable
and results are clear-cut and well-behaved.
Convex set in Rn : S  Rn is a conxex set iff for all x1  S and x2  S, we have:
tx1 + (1-t)x2  S for all t in the interval 0  t  1.
Theorem: Intersection of Convex Sets is Convex.
1.2.3 Relations and Functions
Consider the two sets: set S (s1, s2...) and set T (t1, t2,...)
The product of the two sets: SxT = {(s,t)| s  S and t  T} : Order pairs.
Any Collection or Set of Order pairs is said to constitute a Binary Relation (SxT) of the
sets S and T.
Meaningful Relationship: (sRt) : The Set of the order pairs that are constituted by elements
of the sets S and T are meaningful.
sRt  {( s, t ) | s  S , t  T
and
sRt  S  T }
Example: S  S  S 2  {( x, y) | x  S , y  S} " "  {( x, y ) | x  S , y  S , x  y}
Completeness: A relation R on S is complete if, for all distinct elements x and y in S, xRy or
yRx ( the order pairs (x,y) or (y,x) are all meaningful.)
Reflexivity: A relation R on S is reflexive if, for all elements x in S, xRx (oder pairs (x,x) are
meaningful)
Transitivity: A relation R on S is Transitive if, for all three elements x, y and z in S, xRy and
yRz implies xRz
Odering: A binary relation which satisfies all properties of Completeness, Reflexivity and
Transitivity is called an Ordering.
A Function: Is a Relation that associate each element of one set with a single, unique element
of another set. f: D  R: D is the Domain and R is calledthe Range.
1.3. A little Topology
Topology is a study of fundamental properties of sets and mappings.
The distance between the points: x1 ( x11 , x12 ) and x 2 ( x12 , x 22 ) :
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d(x1, x2) = |x1 – x2| =
( x11  x12 ) 2  ( x12  x 22 ) 2
Open Ball: The open ball with the center x0  R n and radius e > 0 (a real number) is the
subset of poins in R n : Be ( x0 )  {x  R n | d ( x0 , x)  e} .
Closed Ball: the closed ball with center x0  R n and the radius e> 0 (a real number) is the
subset of points in R n : Be ( x0 )  {x  R n | d ( x0 , x)  e} .
Open Sets: S  R n is an Open Set if for all x  S, there exist some e > 0 such that the open
ball Be (x)  S .
The Theorem on Open Sets in R n :
1. The Empty set is an Open Set
2. The entire space R n is an open set
3. The union of open sets is an Open set
4. The intersection of any finite number of open sets is an Open set.
Theorem: Every Open set is a Collection of Open Balls.
S is an open set. For every x  S, choose some e x > 0 such that
Be x ( x )  S .
S   Bex (x)
xS
Closed Sets: S  R n is a closed set if and only if its complement cS is an Open set.
Theorem on Closed sets:
1. The empty set is a closed set
2. The entire space is a closed set
3. The union of closed set is a close set
4. The intersection of a finite number of closed sets is a closed set.
Theorem: Closed sets in R and the Unions of close Intervals.
Let S is any closed set in R. Then:
S=
 (, ai]  [bi,)
iI
Proof:
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If S  R is closed  cS is open. By the definition of an open set, for each x in cS,
we have:
cS   B (x) .
We can rewrite as: cS   ( x   x , x   x ) .
xcS
Let i=x, I=cS, ai  x   x , bi  x   x , we have ai<bi
cS   (ai , bi )
iI
 S  c  (ai , bi )
iI
Applying the De Morgan’s law, we have: S   c(ai , bi )
iI
c(ai , bi )  (, ai ]  [bi ,)
 S   (, ai ]  [bi ,) .
iI
Theorem: Close sets in R and the Union of Closed Intervals:
Let S is any closed set in R . Then:
S=
 [0, ai]  [bi,)
iI
Bounded Sets in R n : A subset S in R n is called bounded if and only if it is entirely
contained within some ball (an open or a closed ball). That is, S is bounded if there exists e>0
such that S  Be (x) for some x  R n .
Consider a subset S in R space:
A lower bound: A real number l is called a lower bound for S if l  x for all x  S .
An Upper bound: A real number l is called an upper bound for S if l  x for all
xS.
A subset S in R has many lower bounds and upper bounds.
The greatest lower bound (g.l.b): the biggest number among those lower bounds for
S.
The least upper bound (l.u.b): the smallest number among those upper bounds for S.
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A close set S in R contains its upper and lower bounds. An open set S in R does not
contain its upper and lower bounds.
Theorem: Upper and Lower Bounds in Subsets of Real Numbers
1. Let S  R is a bounded open set and let a be the g.l.b of S and b be the l.u.b of S.
Then a  S and b  S .
2. Let S is a bounded closed set in R. Let a is the g.l.b of S and b is the l.u.b of S.
Then a  S and b  S.
Compact Sets in R n ( Heine-Borel): A set S  R n is compact if and only if S is closed and
bounded.
R n is closed but is not bounded  R n is not compact.
closed
and
bounded
1.3.1. Continuity
- A Continuous mapping or a continuous function
- In most economic application, we will either want to assume that the function we are dealing
with are continuous or we want to discover whether they are continuous when we are
unwilling to assume it.
- A continuous function: f: R  R is continuous function at a point xo if for all  > 0,
there exists  > 0 such that d(x, xo)<  implies d((f(x)-f(xo)) <  .
f ( B ( xo)  B ( f ( xo))
f(xo+
)
f(xo)
f(xo+  )
Geoffrey A. Jehle – Advanced Microeconomic
xo xo+Theory

f(xo+  )

f(xo)+
f(xo)
xo
xo+ 
f(xo+  )  (f(xo), f(xo)+
)
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Continuity: Let D be a set, R be another set, and let f: D  R. The function f is continuous
at the point xo  D if, and only if, for all
f ( B ( xo))  B ( f ( xo))

> 0, there exists a  > 0 such that:
. If f is continous at all xo  D, it is called a
continuous function.
A function is continuous at a point xo if for all  > 0, there exists  > 0 such that any point
less than a distance  away from xo is mapped by f into some point in the range which is less
than a distance  away from f(xo).
Every point in B (xo) is mapped by f into some point no father than  from f(xo).
f(xo) is the image of xo.
Basically, a function is continuous if a “small movement” in domain does not cause a “big
jump” in the range.
It is not true that a continuous function always maps an open set in the domain set into an
open set in the range, or that closed set is mapped into closed sets. Ex: y = a. (the domain is
an open set but the range (image) set may not be an open set)
Theorem: Continuity and the Inverse Image of Open Sets
Let f: D  R be a mapping and let f
1
: R  D be its inverse mapping from R to D. Let
T  R be an open set in the range of f. Then f is continuous if and only if the inverse
image f 1 (T )  D is an open set in the domain.
Proof:
Necessity: Let f is a continuous function and T be an open set in the range. We have to prove
that f 1 (T )  D is an open set.
T = {f(x)} is an open set   some   0 such that
B ( f ( x))  T . The function f is
a continuous function  according to the definition of a continous function, there exist   0
such that
f ( B ( x))  B ( f ( x))  B ( x)  f 1[ B ( f ( x))]
 f 1 (T )  D is an
open set because f 1 (T )  D is a set of B (x) .
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Sufficiency: Let
f 1 ( B ( f ( x))  D
is an open set. We need to prove that f(x) is a
continuous function.
If
f 1 ( B ( f ( x))  D
contained by f
for all
1
is an open set   some   0 such that B ( x)  D is
( B ( x)) .  B (x ) 
f
1
( B ( x))  f ( B ( x))  B f ( x)
x  { f 1 ( B ( f ( x))} .
 f(x) is a continuous function for all
x  { f 1 ( B ( f ( x))} .
Theorem: Continuity and the Inverse Image of Closed Sets
Let f: D  R is a continuous mapping and f
1
is inverse mapping from R to D. T  R is
closed set in the range of f. Then f is continuous function if and only if f 1 (T )  D is a
closed set in the domain of f.
Proof:
Let an image set T of f(x) is a closed set. We need to prove that f 1 (T )  D is a closed set
equivalent to f being a continuous function.
If T is a closed set  cT is an open set . According to the preceding theorem, f(x) is a
continuous function if and only if f 1 (cT ) is an open set or cf 1 (T ) is an open set 
f 1 (T ) is a closed set.
Theorem: The continuous Image of a Compact Set is a Compact set.
1.3.2. Some Existence Theorems
-
All Existence Theorems specifies conditions if met, guarantee that something exists.
-
The conditions in Existence Theorems are sufficient conditions.
something
E.T.Cs
E.T.Cs do not hold
Something doesn’t hold
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-
While these theorems assure us that something exists, they generally give no clue as to
what it may look like, or where we may find it.
-
Optimization theory
-
The Weierstrass Theorem specifies sufficient conditions under which the existence of
a maximum and a minimum of a continuous function are assured.
Theorem Weierstrass: Existence of Extreme Values
The Weierstrass Theorem specifies sufficient conditions under which the existence of a
maximum and a minimum of a continuous function are assured.
The theorem: Let f: R n  R be a continuous real valued mapping. Let S be a compact
(closed and bounded) subset of R n . Then there exists a vector x   S and a vector
~
x  S such that:
f ( x* )  f ( x)  f ( ~
x)
for all x  S .
Proof:
According to the theorem: The continuous image of a Compact set is a compact set (D is a
compact set, f is continuous  the Range/image of D is a compact set. We have that S is
a compact set and the mapping is continuous  the image of S is a compact set.
Obviously, the image set of S  R   Greatest Lower Bound = a (g.l.b) and Smallest
Upper Bound = b (s.u.b) of the set image of S  there is x1 and x2 in S such that
f(a)=g.l.b and f(b)=s.l.b due to the property of a closed set S  f (a)  f ( x)  f (b) .
.
Max
Min
x1
x2
Hyperplane H in R n : A Hyperplane H in R n is a set of vectors x sastifying ax   for
some a  R n , a  0 and   0 .
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A Hyperplane H separates two sets S and T if: a.x   for all x  S and a.x   for all
x T .
A hyperplance H in R n
The hyperplane H separates S and T if:
a.x   for all x  S
a.x   for all x  T
Theorem: Separating a Point and a Convex Set
Let S in R n be convex, closed and nonempty. Let y 0  R n and y 0  S . Then there exists
some a  R n , a  0 and   R such that all three of the following hold:
1. a.y 0  
2. a.x'   for some x’ is on the boundary of S ( S ).
3. a.x  
(1) and (3) tell that there is some hyperplane that separates the point y0 and the set S. (2)
means that the hyperplane will pass through a point on the boundary of S.
x’ will be the closest point in S to y0.
a = x’ – y0 and a.x’ =  .
Lemma: Under the condition of the preceding theorem, there exists at least one point
x' S (boundary of set S) such that d(y0, x’)  d(y0, x) for all x  S and d(y0, x’)>0.
x2
S
xo
B*
A
yo
x1
Prove the Lemma:
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*
Let xo be any poin in S and let  be ||yo-xo||. Let B be a closed ball centered at yo with
*
*
radius  . Let A be intersection of B and S. Because S and B are closed and nonempty
 the intersection A is closed and nonempty  A is compact. Let A={x’}.
Consider the function d(yo,x’) over domain A. This is a continuous function. Because A
– domain is a compact set  According to Weierstrass theorem, there exists x’ in A such
that d(y0, x’)  d(yo,x) for all x  A . Because x’ is in S and yo is not in S, so d(yo.x’)>0.
So x’ is the closest point in A to yo . It is easy to realize that every point in S that is not in
the closed ball
B* must be a distance from yo which is trictly greater than  . So x’ is the
closest point in S set to yo. x’ must be on the boundary of S, ortherwise that there exists a
ball centered by x’ means there is other point which is closer to yo.
Prove the theorem: Separating a Point and a Convex set
Let a = x’ – yo, where x’ is the closest point in S to yo. a  0, due to x’  yo.
Let  = a.x’
Prove (1) a.yo <  :
a.yo = (x’-yo).yo = (x’-yo)yo +(x’-yo)x’ – (x’-yo)x’ = - (x’-yo)(x’-yo) + (x’-yo)x’
= -||x’-yo||2 +(x’-yo)x’ = -||x’-yo||2 + 
 a.yo -  < 0  a.yo <  .
Prove (2) a.x’ =  :
By the Lemma, x’ must be on the boundary of S and by the definition of  .
Prove (3) a.x   for all x in S.
S is a convex set  xt = tx + (1-t)x’ will be in S. ( 0  t  1)
Due to x’ is the closest point in S to yo  ||x’-yo||  ||xt – yo||  ||x’-yo||2  ||xt – yo||2
Consider ||xt – yo||2 = ||tx + (1-t)x’ –yo||2 = ||(1-t)(x’-yo) + t(x-yo)||2
=((1-t)(x’-yo) + t(x-yo)). ((1-t)(x’-yo) + t(x-yo))
= (1-2t+t2) ||x’-yo|| 2 +2t(1-t)(x’-yo)(x-yo) + t2||x-yo||2
 ||x’-yo||  (1-2t+t2) ||x’-yo|| 2 +2t(1-t)(x’-yo)(x-yo) + t2||x-yo||2
 0  (1-2t+t2) ||x’-yo|| 2 +2t(1-t)(x’-yo)(x-yo) + t2||x-yo||2 - ||x’-yo||
 0  t(t-2) ||x’-yo|| 2 +2t(1-t)(x’-yo)(x-yo) + t2||x-yo||2
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due to t > 0, we can devide both sides by t to get:
 0  (t-2) ||x’-yo|| 2 +2(1-t)(x’-yo)(x-yo) + t||x-yo||2
Hold for all t > 0, it must hold in the limit as t  0:
 0  -2 ||x’-yo|| 2 +2(x’-yo)(x-yo)
 0  -2( (x’-yo). (x’-yo) +(x’-yo)(x-yo) )
 0  (x’-yo).x’ - (x’-yo)x
 (x’-yo).x’  (x’-yo)x  a.x  
Theorem: The Minkowski Separation Theorem
Let S and T be two nonempty, disjoint, and convex sets in R n . Then there is an
a  R n , a  0 and an   R n such that a.x   for all x  S and a.x   for all x  T .
Theorem: The Brouwer Fixed Point Theorem
Let S  R n be a compact and convex set. Let f: S  S be a continuous mapping. Then
there exists at least one fixed point of f in S. That is, there exist at least one x*  S such
that x* = f(x*).
For all x  S we have f(x)  S as well.
S  R n and f: S  S , then f maps vectors in S back into other vector in the same set S.
Linear of Nonlinear systems of equations define functions of this sort.
y1  g 1 ( x1,...xn)
.
.
n
yn  g ( x1,...gn)
maps points (x1,...,xn)  R n into points (y1,...,yn)  R n
A fixed point: (special case)
x1*  f 1 ( x1* ,...xn* )
.
.
xn*  f n ( x1* ,...xn* )
(x1 * ,...,xn * ) is called a fixed point
Prove the Brouwer Fixed Point theorem
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We need to prove that for all x  S , f(x)  S as well, there exists at least x*  S such that
x*=f(x*).
This proof is only restricted in considering mapping: f: R  R
Let S = [a, b]  R is a compact and convex. Because f(x)  S  f(x)  [a,b].
 a  f ( x)  b or a  f (x) and b  f (x) (P.1)
Construct a new function: g ( x)  x  f ( x) (P.2)
and let the domain of g be the same interval [a,b].
y = x is a continuous function, f(x) is assumed to be a continuous function  g(x) is a
continuous function (difference of continuous functions is a continuous function).
Consider g(x) at the endpoints a, b:
We have g(a) = a – f(a)  0 and g(b) = b – f(b)  0 (according to P.2)
If one of these inequalities holds with equality, a or b will be a fixed point: a = f(a) or b=f(b).
Now we must consider the possibility of x = f(x) that g(a)<0 and g(b)>0.
We know g(a) < 0, we always find out c > a such that g(x) < 0 for all x  [a, c). Proving this
statement is rather easy. Beacase g(x) is a continuous function over [a, b]. g(x) is continuous
a
at a if and only if for all   0 , there exists   0 such that g( B )  B
g (a )
0 , we can find out   0 that is small enough such that B
g (a)
. Because g(a) <
 R (negative real number
set).  We can find c > a (maybe c = a+  ) such that g(x) < 0 for all x  [a, c).
Similiarly, we can find c<b such that g(x) > 0 for all x  (c, b] .
Let c* is the largest number to make the open interval [a, c*) such that g(x)<0 for all
x  [a, c*) . Let x* is the least upper bound of [a,c*)  x* = c.
Since x* = c, we know that a < x* < b and g(x) < 0 for all x  [a, x*) .
We realize that g(x*) can not be negative. If g(x*)<0  -g(x*)>0. Because g(x) is continuous
function at x*, we take  = -g(x*) > 0, there exists  >0 such that g ( B ( x*))  B g ( x*) g ( x*) .
B g ( x*) g ( x*)  (0,2 g ( x*)) , this contradics the assumption that g(x*)<0.
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g(x*) can not be positive as well. If g(x*)>0, we take   g (x*) , then there exists  >0 such
that
g ( B ( x*))  Bg ( x*) g ( x*) . And Bg ( x*) g ( x*)  (0,2g ( x*)) . This vilolates the
condition g(x)<0 for all x  [a, x*) .
So, g(x*) can not be both positive and negative  g(x*) = 0  x* = f(x).
In the paragraph, the line f will cross the 450 curve at least one time. So according to
Brouwer’s Theorem, the fixed point is not unique.
1.4.
Real Valued Function
Definition fo Real Valued Function: f: D  R is a real valued function if D is any set and
R  R.
This section we will restrict out attention to real valued functions whose domains are convex
sets.
Assumption: Real Valued Function over Convex Sets.
Let f: D  R is a real valued function where D  R n is a convex set and R  R .
Increasing Functions: f: D  R is increasing function whenever f(x0)  f(x)  x 0  x , and
x 0  x . We say f is strictly increasing function whenever f(x0) >f(x)  x0  x, and x 0  x .
Note: x 0  x means that at least one of components of vector x0 is greater than the same
ordered component of vector x.
Decreasing Functions: f: D  R is decreasing function whenever f(x0)  f(x)  x 0  x ,
and x 0  x . We say f is strictly increasing function whenever f (x0) < f(x)  x 0  x , and
x0  x .
1.4.1. Related Sets
The graph of a function is a related set which provices an easy and intuitive way of thinking
about the function.
There are some sets related to a function.
Level Sets: L(y0) is a level set of the real valued function f: D  R iff
L(y0)  {x | x  D, f(x)  yo} , where y0  R .
Examples: Isoquant curve, Indiferrent curves, Isoprofit curve…
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Using of Level Sets:
-
Reducing by one the number of dimesions needed to represent the function.
-
Two indifferent level sets of a function can never cross
Level Sets Relative to a Point x0
L(x0) is a level sets relative to point x iff L(x0) = {x|x  D , f(x) = f(x0)}.
Superior and Inferior Sets
1. S(yo) = {x | x  D , f(x)  yo} is call superior set for level yo.
2. I(yo) = {x | x  D , f(x)  yo} is call inferior set for level yo.
3. S’(yo) = {x | x  D , f(x)  yo} is call strictly superior set for level yo.
4. I’(yo) = {x | x  D , f(x)  yo} is call strictly inferior set for level yo.
Theorem: Superior, Inferior, and Level Sets
For any f: D  R any y 0  R :
1. L(yo)  S(yo)
2. L(yo)  I(yo)
3. L(yo) = S(yo)  I(yo)
4. S’(yo)  S(yo)
5. I’(yo)  I(yo)
6. S’(yo)  L(yo) = O
7. I’(yo)  L(yo) = O
8. S’(yo)  I’(yo) = O.
Concave Functions: f: D  R is a concave function if and only if
f ( xt )  tf ( x1)  (1  t ) f ( x2)
for all t  [0,1] .
(f(x) is a real valued fucntion)
-
The set of points beneath concave regions is a convex set. The set of points beneath
the non-concave region is not a convex set.
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y
x
Theorem: Points On and Below the Graph of a Concave Function Always Form a
Convex Set
Let D  R n be a convex set and let R  R. Let A  {( x, y ) | x  D, f ( x)  y} be the set of
points “on or below” the graph of f: D  R . Then f is concave function  A is a convex
set.
Prove: We have to show that f(x) is a concave function implies A is convex and A is
convex implies f(x) is concave function.
First part: f(x) is a concave function  A is a convex set.
Let take any two points: (x1,y1) and (x2,y2) in set A
Take convex combination of the two point: (xt, yt) such that:
xt = tx1 + (1-t)x2 and yt = ty1 + (1-t)y2 .
We need to prove that (xt,yt) is also in set A.
F(x) is a concave function  by the definition of concave function:
f(xt)  tf(x1) + (1-t)f(x2)
By the definition of set A, we have f(x1)  y1 and f(x2)  y2
 tf(x1) + (1-t)f(x2)  ty1 + (1-t)y2 = yt
 f(xt)  yt  (xt,yt)  A  A is a convex set.
Second part: Prove A is convex implies f is a concave set.
Consider two any points on the graph of f(x): (x1,f(x1)) and (x2,f(x2)).
A is convex  (xt, yt)  A, in which :
Xt = tx1 + (1-t)x2 and yt = tf(x1) + (1-t)f(x2).
We need to prove that: f(xt)  tf(x1) + (1-t)f(x2).
According to the definition of set A: f(xt)  yt = tf(x1) + (1-t)f(x2).
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 f(x) is a concave function.
Strictly Concave Functions
f: D  R is a strictly concave function iff for all x1 and x2 in D,
f ( xt )  tf ( x1)  (1  t ) f ( x2) for all t  (0,1)
Geometrically, these modification simply require the graph of the function to lie
everwhere strictly above the chord connecting any two points on the graph.
Rule out the flat portions on the graph of the function.
y
x
1.4.3. Quansiconcave Functions
Quansiconcave functions:
f: D  R is quansiconcave function for all x1 and x2 in D, f ( xt )  min[ f ( x1 ), f ( x2 )]
for all t  [0,1].
Geometrically:
-
When f(x) is an increasing function, it will be quansiconcave whenever the level set
relative to any convex combination of two points, L(xt) is always on or above the
lowest of the level sets L(x1) and L(x2).
x2
x1
xt
x2
x1
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-
When f(x) is a decreasing function, it will be quansiconcave whenever the level set
relative to any convex combination of two points, L(xt) is always on or below the
highest of the level sets L(x1), L(x2).
x2
x2
xt
x1
x1
Theorem: Quansiconcave and the Superior Sets
f: D  R is a quansiconcave function if and only if S(x) –( the superior set relative to point
x: {x| x  D , f  f(x)}) is a convex set for all x  D .
Prove: We have to prove the both necessary and sufficient terms.
Sufficiency: Prove that f(x) is a quansiconcave function  S(x) is a convex set.
Consider any two point x1 and x2 in set S(x). We need to prove that xt made by the convex
combination of the two vectors x1 and x2 is also in S(x)  f(xt)  f(x).
 according to the definition of S(x), we have: f(x1)  f(x) and f(x2)  f(x).
According to the definition of a quansiconcave function:  f(xt)  min[f(x1),f(x2)]  f(x).
 f(xt)  f(x).
Necessarity: Prove that if S(x) is a convex set  f(x) is a quansiconcave function.
We need to prove that f(xt)  min[f(x),f(x2)] under the condition that S(x) is convex.
Consider any two points x1 and x2 in S(x). Without loss of generality, assume we have:
f(x1)  f(x2).
According to the definition of S(x), we have: f(x1)  f(x2)  f(x).
Because f(x1)  f(x2)  S(x2)  S(x1)  x1 and x2 are both in S(x2)
 xt = tx1 + (1-t)x2 (for all t  [0,1] ) is also in S(x2)  f(xt)  f(x1)  f(x2).
Because t  [0,1]  f(xt)  min[f(x1), f(x2)]  f(x) is a quansiconcave function.
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x2
x2
Increasing function
Decreasing function
S(x)
S(x)
L(x)
L(x)
x1
x1
Strictly Concave Functions:
A funtion f: D  R is strictly concave function iff, for all x1  x2 in D,
f(xt) > min[ f(x1) , f(x2) ] for all t  (0,1).
-
A Strictly Concave Function forbid the convex combination of two points in the same
level set also lies in that level set
x2
x2
Not Strictly Quansiconcave
x2
Increasing Function
xt
x1
xt
x2
L(x1)=L(x2)=L(xt)
x1
x1
x1
Strictly Quansiconcave Function
Theorem: Concavity Implies Quansiconcavity (Concavity  Quansiconcavity)
A concave function is always quansiconcave. A strictly concave function is always strictly
quansiconcave.
Prove: If f is a concave function, f is always a quansiconcave function
Consider x1 and x2 in D. Without loss of generality, we assume that f(x1)  f(x2).
f(x) is a concave function  f(xt)  tf(xt) + (1-t)f(x2) = f(x2) + t[f(x1) – f(x2)].
Because f(x1) – f(x2)  0  f(xt)  f(x2).
f(x1)  f(x2)  f(x2) = min[f(x1),f(x2)]  f(xt)  min[f(x1),f(x2)]  f(x) is a
quansiconcave function.
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1.4.4 Convex and Quansiconvex Functions
Convex Functions:
f: D  R is a convex function iff for all x1 and x2 in D, f(xt)  tf(x1) + (1-t)f(x2),
for all t  [0,1].
Convexity  the region above the graph – set (x,y) or (D,R) is a convex set.
y
x
Strictly Convex Function:
f: D  R is a strictly convex function iff for all x1 and x2 in D, f(xt) < tf(x1) + (1-t)f(x2),
for all t  (0,1).
Theorem. Points On and Above the Graph of a Convex Function Always Form a Convex
Set
Let D  R n be a convex set, R  R. Let A* = {(x,y) | x  D, f(x)  y} be the set of points
“on and above” the graph of f: D  R.
f is a convex function  A* is a convex set.
Prove:
Sufficiency: We need to prove that f is a convex function  A* is a convex set.
Consider the two points (x1,y1) and (x2,y2) in set A*.
Take the convex combination of the two points, we have: (xt,yt), in which:
xt = tx1 + (1-t)x2 and yt = tf(x1) + (1-t)f(x2)
We need to prove that (xt,yt) is also in set A*  f(xt)  yt.
According to the definition of a convex function: f(xt)  tf(x1) + (1-t)f(x2) = yt
 f(xt)  yt  (xt,yt) is in set A* as well.
Necessity: We need to prove that A* is a convex set  f is a convex function.
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Consider the two points on the graph of f (x): (x1,f(x1) and (x2,f(x2)).
A* is a convex set  f(xt)  tf(x1) + (1-t)f(x2)  is a convex function.
Quansiconvex and Strictly Quansiconvex Functions
1. A function f: D  R is Quansiconvex iff for all x1 and x2 in D,
f(xt)  max[f(x1),f(x2)] for all t  [0,1].
2. A function f: D  R is Strictly Quansiconvex iff for all x1 and x2 in D,
f(xt) < max [f(x1),f(x2)] for all t  (0,1).
Theorem: Quansiconvexity and the Inferior Sets
f: D  R is a quansiconvex function iff I(x) – Inferior set is a convex set for all x  D.
x2
x2
Increasing function
Decreasing function
Inferior set
Inferior set
x1
x1
Prove:
Sufficiency: We need to prove that f(x) is a quansiconvex  I(x) Inferior set is a convex
set.
We need to prove that (xt,yt) is also in the I(x) set.
Consider the two points in I(x) set (x1,y1) and (x2,y2). Without loss of generality, we
assume that f(x1)  f(x2)  I(x1)  I(x2) and f(x1)-f(x2)  0
F(x) is quansiconvex  f(xt)  max[f(x1),f(x2)]
Because f(x1)  f(x2)  f(x2) = max[f(x1),f(x2)]
 f(xt)  f(x2)  (xt,yt)  I(x2)  (xt,yt)  A*.
Necessarity: We need to prove that I(x) is a convex set  f(x) is a quansiconvex
function.
We need to prove that f(xt)  max[f(x1),f(x2)].
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Consider the two points in I(x): x1(x11,x12) and x2(x22,x22)
I(x) is a convex set  the convex combination of the two points is also in I(x).
Without loss of generaliy, we assume that f(x1)  f(x2)  max[f(x1),f(x2)]=f(x2).
f(xt) = tf(x1) + (1-t)f(x2) = f(x2) + t[f(x1)-f(x2)]  f(x2) due to f(x1)  f(x2).
 f(xt)  max[f(x1),f(x2)]  f(x) is a quansiconvex function.
Theorem: Concave/Convex and Quansiconcave/Quansiconvex Funtions
1. f(x) is (strictly) concave function iff –f(x) is (strictly) convex function.
2. f(x) is (strictly) quansiconcave function iff –f(x) is (strictly) quansiconvex function.
Summaries:
1. f concave  convex sets beneath the graph.
2. f convex  convex sets above the graph.
3. f quansiconcave  Superior sets are convex sets.
4. f quansiconvex  Inferior sets are convex sets.
5. f concave  f quansiconcave.
6. f convex  f quansiconvex.
7. f (strictly) concave  -f (strictly) convex.
8. f (strictly) quansiconcave  -f (strictly) quansiconvex.
Strictly Quansiconvex function
y
y
f(x2)
f(x1)
f(xt)
X
X1
Xt
X2
f(xt) < min[f(x1),f(x2)] < max[f(x1),f(x2)]
Geoffrey A. Jehle – Advanced Microeconomic Theory
x
Inferior set is a convex set
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Strictly Quansiconcave function
f(xt)
y
f(x2)
f(x1)
x
X1
Xt
X2
Superior set is a convex set
f(xt) > max[f(x1),f(x2)] > min[f(x1),f(x2)]
f(xt)
f(x1)
f(x1)
f(xt)
f(x2)
f(x2)
x1
x1
xt
xt
x2
x2
f(x) is concave  f(x) is
quansiconcave:
f ( xt)  min( f ( x1), f ( x 2))
Geoffrey A. Jehle – Advanced Microeconomic Theory
f ( xt)  min( f ( x1), f ( x 2))
But f(x) is not concave function
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