Economics

DO 14-8 RD (point function) (set function) Figure 1.5 Point versus set function. real-valued function whose domain-consists of a collection of points, where points are represented by coordinate vectors in E". We have encountered examples of this type of function previously in Example A.15 through Example A.17. A typical ordered pair associated with a real-valued point function is of the form (x,y), where x is a vector in R" and y is a real number in R. A set function is more general than a point function in that its domain consists of a collection of sets rather than a collection of points. A typical ordered pair belonging to a real-valued set function would have the form (Ayl, where A is a set of some type of objects and y is a real number in R. If the sets in the domain of the set function are contained in R", i.e., they are sets of real numbers, then a real-valued set function is assigns a real number to each set of points in its domain in contrast to a real-valued point function which would assign a real number to each point in its domain. A pictorial illustration of a set function contrasted with a point function is given in Figure A.5. Examples of set functions are presented below. {A1, A2,..., As), where A1 = {1},A2 = {2},A3 = {3}, A4 = {1,2},A5 = {1,3), A6 = {2,3), The following is a real-valued set function f: A-R: = {(2,3):= {x, ACA} where x x signifies the sum of the numerical values of all of the elements in the set As, and cox is defined to be zero. The range of the set funetion is