that the images of distinct elements of X under the function f are distinct, but the image of two distinct elements and of X under f is same, namely b . Further, there are some elements like e and f in X which are not images of any element of X under f , while all elements of X are images of some elements of X under f . The above observations lead to the following definitions: Definition A function f : X → Y is defined to be one-one (or injective ), if the images of distinct elements of X under f are distinct, i.e., for every x , x ∈ X, f ( x ) = f ( x ) implies x = x . Otherwise, f is called many-one .
The function f and f in Fig . (i) and (iv) are one-one and the function f and f in Fig . (ii) and (iii) are many-one. Definition A function f : X → Y is said to be onto (or surjective ), if every element of Y is the image of some element of X under f , i.e., for every y ∈ Y, there exists an element x in X such that f ( x ) = y .
The function f and f in Fig . (iii), (iv) are onto and the function f in Fig . (i) is not onto as elements e , f in X are not the image of any element in X under f . Remark f : X → Y is onto if and only if Range of f = Y.
Definition A function f : X → Y is said to be one-one and onto (or bijective ), if f is both one-one and onto. The function f in Fig . (iv) is one-one and onto. Example Let A be the set of