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applications of set theory in economical problem

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applications of set theory in economical problem

  2. 2. Presented By: Nafisha Tasnim Zaman –B1506150 Shamin Yeaser Rahman –B1506131 Sarker humaira mostareen –B1506014 Farjana Islam Mim –B1506182
  3. 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. 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. 5. BASIC CONCEPTS OF SET THEORY  Union  Intersection  Set difference  Symmetric difference  Cartesian product  Power set
  6. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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