Page 12 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Note that the sets are disjoint or mutually exclusive when S  T = 0 i.e., when their intersection
is empty.

You can now verify (or even prove) by means of examples the following laws of union and
intersection of sets given in the next exercise.
Also, you can easily see that

A  A = A, A  A = A, A  = A, A   = .
Given any two sets S and T, we can construct a new set in such a way that it contains only those
elements of one of the sets which do not belong to the other. Such a set is called the difference of
the given sets. There will be two such sets denoted by S—T and T—S. For example, let

S = {2, 4, 8, 10, 11}, T = {1, 2, 3.4).
Then
S – T = {8, 10, 11), T – S = {1, 3).

Thus, we can define the difference of two sets in the following way.
Definition 5: Difference of two Sets
Given two sets S and T, the difference S – T is a set consisting of precisely those members of S
which are not in T.
Thus

S – T = {x:x  S and x  T).
Similarly, we can define T – S.
Consider a collection of sets S , where i varies over some index set J. This simply means that to
1
each element i  J, there is a corresponding set S . For example, the collection {S , S , S ,...) could
i 1 2 3
be expressed as {S }  , where N is the index set.
i i N
With the introduction of an index set, the notions of the union and the intersection of sets can be
extended to an arbitrary collection of sets. For example,


(i)  S =  {x : x S for at least one i  J}.
i
i


i J i J

(ii)  S =  {x : x S for all i  J}.
i
i

i J i J

c

(iii) ( S ) c  =  S .
i
i i J
i J

1.1.2 Functions
Let S be the set of Excel Books and let N be the set of all natural numbers. Assign to each book the
number of pages the book contains. Here each book corresponds to a unique natural number. In
other words, there is a correspondence between the books and the natural numbers, i.e., there is
a rule or a mechanism by which we can associate to each book one and only one natural number.
Such a rule or correspondence is named as a function or a mapping.
Definition 6: Function
Let S and T be any two non-empty sets. A function f from S to T denoted as f: ST is a rule which
assigns to each element of the set S, a unique element in the set T.

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