Set theory is widely applied in economics. It includes concepts like union, intersection, and difference that are used in functions and formulas. Fixed point theorems and Sperner's lemma use set theory to determine if solutions exist to systems of equations, which is important for general equilibrium analysis. While limitations exist in some theoretical economic answers, set theory provides insights and most economists still find it useful as a framework without the problems necessarily affecting their work. It offers a way to model choices, equilibriums, and states without regard for difficulties in the foundations of economic theory.

1. APPLICATIONS OF SET THEORY IN
THE FIELD OF ECONOMICS

2. Presented By:
Nafisha Tasnim Zaman –B1506150
Shamin Yeaser Rahman –B1506131
Sarker humaira mostareen –B1506014
Farjana Islam Mim –B1506182

3. WHAT IS SET THEORY ?
 Set theory is the branch of mathematical logic that
studies sets, which are collections of objects.
Although any type of object can be collected into a
set, set theory is applied most often to objects that
are relevant to mathematics. The language of set
theory can be used in the definitions of nearly all
mathematical objects.

4. AREA OF STUDY :
 1. Combinatorial set theory
 2.Descriptive set theory
 3.Fuzzy set theory
 4.Inner model theory
 5.Determinacy
 6.Forcing
 7.Cardinal invariants

5. BASIC CONCEPTS OF SET THEORY
 Union
 Intersection
 Set difference
 Symmetric difference
 Cartesian product
 Power set

6. FORMULAS OF SET THEORY :
 Identity Laws:
 for any set A, there is one and only one set , such that
 (ii) for any set A, there is one and only one set S, such that .
 Closure Laws:
 If A and B are two sets, then
 (i) is a set,
 (ii) is a set.
 Commutative Laws:
 If A and B are two sets, then
 (i) ,
 (ii)
 A =A
A S =A
A B
A B
A B =B A 
A B =B A 

7. FORMULAS OF SET THEORY :
 Associative Laws:
 If A, B and C, are three sets, then
 (i)
 (ii)
 Distributive Laws:
 If A, B and C, are three sets, then
 (i)
 (ii)
 Complementary Laws:
 For every set A, there is one and only one set such that
 (i)
 (ii)
(A B) C =A (B C)   
(A B) C =A (B C)   
A (B C) =(A B) (A C)    
A (B C) =(A B) (A C)    
c
A A = S
c
A A = 

8. FORMULAS OF SET THEORY :
 Difference Laws:
If A, B and C, are three sets, then
 (i)
 (ii)
 (iii)
 (iv)
 (v)
 (vi)
 (vi)
c
A-B =A B
A - B = A - (A B) =(A B) - B 
A - ( B - C ) = (A - B ) (A - C)
(A B ) - C = (A -C) (B - C) 
A - (B C)=(A-B) (A-C) 
(A B ) (A - B) = A 
(A B ) ( A - B ) = 

9. INTRODUCTION OF SET THEORY IN
ECONOMIC !
 The set-theoretical framework we think is better fit for
economics is AFA−+ AD + DC. That is, the theory of sets
that can be derived from the axioms of 2 Zermelo-
Frenkel set theory as well as from the axiom of
Dependent Choices and the Axiom of Determinacy
We claim that a lot of insight is gained in this switch
while no important results for economics are lost.
.

10. FUNCTIONS
 In theoretical economics, such as general equilibrium
analysis, there comes a point where one needs to know
whether the solution to a system of equations exists; or, more
specifically, under which conditions will a solution necessarily
exist.
 Fixed Point Theorems
 No Retraction Theorems
 Sperner's Lemma (One dimensional version)
 Two Dimensional Version

11. Fixed Point Theorems
 Brouwer's Fixed Point Theorem: A continuous
mapping of a convex, closed set into itself
necessarily has a fixed point.
Examples:
 A continous function that maps [0,1] into itself has
a fixed point.
 A continuous function that maps a unit disk into
itself has a fixed point.

12. No Retraction Theorem:
nuous mapping of all points of the interior of disk onto its
boundary circle.
 Proof of Brouwer's fixed point theorem for a disk using the
No Retraction Theorem.
 Assume there is not fixed point and use the intersection of
the line from x to f(x) with the boundary circle to map x
into the boundary. This would be a continuous mapping of
the interior onto the boundary. This is a contradiction of
the No Retraction Theorem so the process must break
down some where. It breaks down if f(x)=x because there is
no unique line defined.

13. Sperner's Lemma (One dimensional
version)
 Consider a line segment AB subdivided into segments
and the end points of the segments labeled with A's and
B's arbitrarily. Let a be the number of segments labeled
AA and b the number of segments labeled AB, complete
segments.
The number of end points labeled with A is 2a+b. Let c be
the number of internal end points labeled A. If we count A
end points segment by segment we get 2c+1. Therefore
2a+b=2c+1, which implies that b=2(c-a)+1 so b, the
number of segments labeled AB, is an odd number. Since
zero is not an odd number there has to be at least one
segment labeled AB.

14. Two dimensional version.
 Let a be the number of triangles whose labels read ABA or
BAB. Triangles of these two types have two edges labeled AB
where as complete triangles have one edge labeled AB. The
other types of triangles have no edges labeled AB.
The number of AB edges counted triangle by triangle is 2a+b.
However, edges inside the original triangle are counted twice
since they belong to two triangles. Let c be the number of edges
labeled AB inside the original triangle.
Let d be the number of edges labeled AB on the outside of the
original triangle. Then the number of AB edges counted is 2c+d
and this is equal to 2a+b. So 2c+d=2a+b. From the preceding
result b must be odd so d also must be odd. Therefore there
must be at least one triangle with labeling ABC.

15. CONCLUSION
we presented a variety of limitations to the theoretical
answers that have been given to the main problems in
economics.
It is not clear that they will (or should) affect the work of
most economists. Although the existence of holes in the
edifice of economic theory is somewhat worrisome, it is
also true that none of the problems discussed here
amounts to its demolition.
In fact, most economists use the existence of choice
functions, equilibria and states of the world as just
metaphors and make models in which the difficulties
discussed here are disregarded