Page 25 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

which gives a verbal description of the elements or a property that is common to all the Notes
elements of a set.

 A set with a limited number of elements is a Finite set. A set with an unlimited number of
elements is an infinite set. A set with no elements is a null-set. A set S is a subset of a set T
if every element of S is in T. The set S is said to be a proper subset of T if every element of
S is in T and there is at-least one element of T which does not belong to S. The sets S and T
are equal if S is a subset of T and T is a subset of S. The null set is a subset of every set and
every set is a subset of itself.

 The union of two sets S and T, written as S  T, includes elements of S and T without
repetitions. The intersection of S and T, written as S n T, includes all those elements that are
c
common to both S and T. The complement of a set S in a Universal set X is denoted as S and
it consists of all those elements of X which do not belong to S. The laws with respect to
union, intersection and complement have been asked in the form of exercises. Also, these
notions have been extended to an arbitrary family of sets.
 A function f: ST is a rule by which you can associate to each element of S, a unique
element of T. The set S is the domain and the set T is the co-domain of f. The set {f(x):x  S}
is the Range of f, where f(x) is an image of x under f. The function f is one-one if f(x ) = f(x )
1 2
 x, = x, for any x, x, in the domain of f. It is said to be onto if the range of f is equal to the
domain of f. A function f is said to be a one-one correspondence if it is both one-one and
onto. A function i: SS defined by i(x) = x,  x  S is called an identity function, while a
function f; ST is said to be constant if f(x) = c, x  S, c being a fixed element of T.

 Any two functions with the same domain are said to be equal if they have the same image
for each point of the domain. The composite of the functions f: ST and g: TV is a
function denoted as ‘g o f’: SV and defined by (g o f) (x) = g(f(x)). The function f: ST is
said to be invertible if there exists a function g: TS such that both ‘g o f’ and ‘f o g’ are
identity functions. Also, a function is invertible if it is both one-one and onto. The inverse
of f exists if f is invertible and it is denoted as f.

 We have discussed the development of the system of numbers starting from the set of
natural numbers. These are the following:

Natural Numbers (Positive Integers):
N = {1, 2, 4 ....}
Integers:
Z = {.... 3, –2, –1, 0, 1, 2, 3 ....}

Rational Numbers:
p
Q = { : p  z, q  Z, q  0}
q
Real Numbers:

R = Disjoint Union of Rational and Irrational Numbers
R = Q  I, Q  I = 
Complex Numbers:

C = {z = x + iy : x  R, y  R}, i =  1.

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