Page 13 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

The set S is called the domain of the function f and T is called its co-domain. If an elements x in Notes
S corresponds to an element y in T under the function f, then y is called the image of x under f.
This is expressed by writing y = f (x). The set {f(x): x  S} which is a subset of T is called the range
of f. If range of f = co-domain of f, then f is called onto or surjective function; otherwise f is called
an into function.
Thus, a function f: ST is said to be onto if the range of S is equal to its co-domain T.

Suppose S = {1, 2, 3, 4) and T = {1, 2, 3, 4, 5, 6) and f: ST is defined by f(n) = n+l,  n  S. Then the
range of f = {2, 3, 4, 5). This shows that f is an into function. On the other hand, if S = {1, 2, 3, 4},
Z
T = {1, 4, 9, 16} and if f: ST is defined by f(n) = n , then f is onto. You can verify that the range of
f is, in fact, equal to T.
Refer back to the example on the books in Excel Books. It just possible that two books may have
the same number of pages. If it is so, then under this function, two different books shall have the
same natural number as their image. However if for a function any two distinct elements in the
domain have distinct images in the co-domain, then the function is called one-one or injective.
Thus a function f is said to be one-one if distinct elements in the domain of f have distinct image
or in other words, if f(x ) = f(x )  x = x , for any x , x in the domain of f.
1 2 1 2 1 2
A function which is one-one and onto, is called a bijection or a 1-1 correspondence.

Example:
(i) Let S = {1, 2, 3) and T = {a, b, c} and let f: ST be defined as f(1) = a, f(2) = b, f(3) = c. Then f
is one-one and onto.
(ii) Let N = {1, 2, 3, 4,...} and f: NN be defined as f(n) = n+1. As 1 does not belong to the range
of f, therefore f is not onto. However, f is one-one.
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(iii) Let S = (1, –1, 2, 3, –3) and let T = (1, 4, 9). Define f: ST by f(n) = n  n  S. Then f is not
one-one as f(1) = f(–l) = 1. However, f is onto.
Definition 7: Identity Function
Let S be any non-empty set. A function f: SS defined by f(x) = x for each x in S is called the
identity function.
It is generally denoted by I . It is easy to see that I is one-one and onto.
s s
Definition 8: Constant Function
Let S and T be any two non-empty sets. A function f: ST defined by f(x) = c, for each x in S, where
c is fixed element of T, is called a constant function.

For example f: SR defined as f(x)=2, for every x in S, is a constant function. Is this function one-
one and onto? Verify it.

Definition 9: Equality of Functions
Any two functions with the same domain are said to be equal if for each point of their domain, they
have the same image. Thus if f and g are any two functions defined on an non-empty set S, then
f = g if f(x) = g(x),  x  S.
In other words, f = g if f and g are identical.

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