1.5 Functions

. Functions Among several relations that exist between two non-empty sets, some special relations are important for further exploration. Such relations are called “ Functions ”. Illustration A company has employees in different categories. If we consider their salary distribution for a month as shown by arrow diagram in Fig. . , we see that there is only one salary associated for every employee of the company. Here are various real life situations illustrating some special relations: . Consider the set A of all of your classmates; corresponding to each student, there is only one age. . You go to a shop to buy a book. If you take out a book, there is only one price corresponding to it; it does not have two prices corresponding to it. (of course, many books may have the same price). . You are aware of Boyle’s law. Corresponding to a given value of pressure P , there is only one value of volume V . . In Economics, the quantity demanded can be expressed as Q P , where P is the price of the commodity. We see that for each value of P , there is only one value of Q . Thus the quantity demanded Q depend on the price P of the commodity. We often come across certain relations, in which, for each element of a set A , there is only one corresponding element of a set B . Such relations are called functions . We usually use the symbol f to denote a functional relation. Definition A relation f between two non-empty sets X and Y is called a function from X to Y if, for each x X Î there exists only one y Î such that ( , ) f Î That is, f ={( x , y )| for all x ∈ X , y ∈ Y }. A function f from X to Y is written as f X : ® Comparing the definitions of relation and function, we see that every function is a relation. Thus, functions are subsets of relations and relations are subsets of cartesian product. (Fig. . (a)) ` 20000 ` 30000 ` 45000 ` 50000 ` 100000 E E E E E Fig. . Employees Salary Fig. . (a) Relation Function Cartesian product Fig. . (b) Inputs Outputs Function Machine Relations and Functions A function f can be thought as a mechanism (or device) (Fig. . ( b )), which gives a unique output f ( x ) to every input x .