Section 1.3: Functions

Consider the functions \(f_1, f_2, f_3\) and \(f_4\) given by the following diagrams.

In Fig 1.2, we observe that the images of distinct elements of \(X_1\) under the function \(f_1\) are distinct, but the image of two distinct elements 1 and 2 of \(X_1\) under \(f_1\) is same, namely \(b\). Further, there are some elements like \(e\) and \(f\) in \(X_2\) which are not images of any element of \(X_1\) under \(f_1\), while all elements of \(X_3\) are images of some elements of \(X_1\) under \(f_3\). The above observations lead to the following definitions:

Definition: Injdective Functions or One-One Functions
A function \(f : X \to Y\) is defined to be one-one (or injective), if the images of distinct elements of \(X\) under \(f\) are distinct, i.e. for every \(x_1, x_2 \in X\), \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\). Otherwise, \(f\) is called many-one.

The functions \(f_1\) and \(f_4\) in Fig 1.2 (i) and (iv) are one-one and the function \(f_2\) and \(f_3\) in Fig 1.2 (ii) and (iii) are many-one.

Definition: Onto Functions or Surjective Functions
A function \(f : X \to Y\) 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 \(y \in Y\), there exists an element \(x\) in \(X\) such that \(f(x) = y\).

The function \(f_3\) and \(f_4\) in Fig 1.2 (iii), (iv) are onto and the function \(f_1\) in Fig 1.2 (i) is not onto as elements \(e, f\) in \(X_2\) are not the image of any element in \(X_1\) under \(f_1\).

Remark
\(f : X \to Y\) is onto if and only if \(\text{Range of } f = Y\).

Definition: Bijective functions
A function \(f : X \to Y\) is said to be one-one and onto (or bijective), if \(f\) is both one-one and onto.

The function \(f_4\) in Fig 1.2 (iv) is one-one and onto.