see that the relation in Example is not a function because the element has no image. Again, the relation in Example is not a function because the elements in the domain are connected to more than one images. Similarly, the relation in Example is also not a function. ( Why ?) In the examples given below, we will see many more relations some of which are functions and others are not.
Example Let N be the set of natural numbers and the relation R be defined on N such that R = {( x , y ) : y = x, x, y ∈ N }. What is the domain, codomain and range of R? Is this relation a function? Solution The domain of R is the set of natural numbers N .
The codomain is also N . The range is the set of even natural numbers. Since every natural number n has one and only one image, this relation is a function. Example Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?
R = {( , ),( , ), ( , )}, (ii) R = {( , ),( , ),( , ), ( , )} (iii) R = {( , ),( , ),( , ), ( , ), ( , ), ( , )} Solution (i) Since , , are the elements of domain of R having their unique images, this relation R is a function. Since the same first element corresponds to two different images and , this relation is not a function. (iii) Since every element has one and only one image, this relation is a function. Definition A function which has either R or one of its subsets as its range is called a real valued function .
Further, if its domain is also either R or a subset of R, it is called a real function . Example Let N be the set of natural numbers. Define a real valued function f : N à N by f ( x ) = x + . Using this definition, complete the table given below.