Page 11 - DMTH401_REAL ANALYSIS

Unit 1: Sets and Numbers

Now, suppose that the Universal set X is given as Notes
X = {1, 2, 3, 4, }
and
S = {1, 2, 3}
is a subset of X. Consider a subset of X whose elements do not belong to S. This set is (4, 2}.

Such, a set, as you know is called the complement of S.
We define the complement of a set as follows:
Definition 2: Complement of a Set
Let X be the Universal set and S be a subset of X. The complement of the set S is the set of all those
elements of the Universal set X which do not belong to S. It is denoted by S.
Thus, if S is an arbitrary set contained in the Universal Set X, then the complement of S is the set
S = {x:x S}.
c
Associated with each set S is the Power set P(S) of S consisting of all the subsets of S. It is written
as
P(S) = {A : A  S).
Now try the following exercise.
Let us consider the sets S and T given as
S = {1, 2, 3, 4, 5}, T = {3, 4, 5, 6, 7}.

Construct a new set {1, 2, 3, 4, 5, 6, 7). Note that all the elements of this set have been taken from
S or T such that no element of S and T is left out. This new set is called the union of the sets S and
T and is denoted by S  T.

Thus
S  T = {1, 2, 3, 4, 5, 6, 7).
Again let us construct another set {3, 4, 5). This set consists of the elements that are common to
both S and T i.e. a set whose elements are in both S and T. This set is called the intersection of S
and T. It is denoted by S  T. Thus
S  T = {3, 4, 5).
These notions of Union and Intersection of ‘two sets’ can be generalized for any sets in the
following way: Note, all the sets under discussion will be treated as subsets of the Universal set
X.
Definition 3: Union of Sets

Let S and T be given sets. The collection of all elements which belong to S or T is called the union
of S and T. It is expressed as
S  T = {x : x S or x T}.
Note that when we say that x  S or x  T, then it means that x belong to S or x belong to T or x
belong to both S and T.
Definition 4: Intersection of Sets
The intersection S  T of the sets S and T is defined to be the set of all those elements which
belong to both S and T i.e.

S  T = {x : x S and x T}.

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