M2L2
Set Theory and Event Operations
1. Introduction
This lecture is a continuation of discussion on random events that started with definition of
various terms related to random events and concepts of probability in previous lecture. Here
Set Theory, its properties and various set or event operations are explained in detail with
relevant examples.
2. Set Theory
To understand Set Theory one should know the definition of ‘Set’ first. A Set is a collection
of physical or mathematical objects called as members or elements. The elements of a set do
not require to have any similarity among them except belong to the same set.
Set Theory is the theory that is concerned with the properties of a set which are independent
of particular elements belong to the set.
For example, i. When a coin is tossed twice, the possible four outcomes or events
and
form the sample space, i.e.
and T/T}. Any
one or more than one possibility from S will form a set. ii. If an experiment is carried out to
determine the strength of concrete for a number of concrete blocks, the all the experiment
results will form a set. iii. If the stage height is recorded at a river gauging station, all the
possible values of stage-height will form a set.
3. Set Properties
The properties of Set Theory should be explored to understand and apply Set Theory in
practical problems. Common properties are explained here with diagram.
Property 1. Two different sets
and
are equal if all elements of the set ‘ ’ belong to set
‘ ’ and vice-versa. In diagram below (Fig. 1), ‘ ’ and ‘ ’ are two sets for which elements are
completely overlapping each other.
Fig. 1. Equal Sets
Property 2. A set ‘ ’ is proper subset of another set ‘ ’ when all elements of set ‘ ’ belongs
to ‘ ’, not vice-versa, it is denoted as:
. In diagram below (Fig. 2) all the elements of
set, ‘ ’ are enclosed by the boundary of set ‘ ’. However, the reverse is not true.
Fig. 2. Subset
Property 3. An empty or null set is defined as the set contains no elements, it is denoted
as: . The diagram below (Fig. 3) shows a set ‘ ’ with no elements within its boundary.
S
Fig. 3. Null Set
Property 4. A set is said to be ordered if a relation (‘ ’ or ‘ ’) for any two elements, so that
a. either
b. if
or
is possible.
and b  c , then
.
4. Set Operations
Several operations on sets or subset or events are possible, which are important in any
analysis using random variables.
4.1.
Union
Union is an event operation when a set of elements are selected based on their occurrence in
two sets ‘ ’ and ‘ ’ (belongs to set ‘ ’) following three alternative conditions: ‘ alone’, ‘
alone’, or both
and . This operation is denoted by:
(read as
union ). The figure
below (Fig. 4) shows that elements shaded in red denoted as ‘ ’ union ‘ ’ that consists of ‘
alone’ or in ‘ alone’, or in the overlapped area.
Fig. 4. Venn diagram for Union
4.2.
Intersection
Intersection is an event operation when a set of elements are selected based on their
occurrence in both the sets - ‘ ’ and ‘ ’ (belongs to set ‘ ’). This is denoted by:
(read
as
intersection ). In figure, the overlapped area (shaded in yellow) of sets ‘ ’ and ‘ ’ are
known as
.
Fig. 5. Venn diagram for Intersection
4.3.
Mutually Exclusive Sets
Two events ‘ ’ and ‘ ’ in set ‘ ’ are mutually exclusive when none of the outcomes in ‘ ’
belongs to ‘ ’ or vice-versa. These are denoted as:
. Venn diagram for such
condition is presented in fig. 6. There is no overlap between ‘ ’ and ‘ ’.
Fig. 6. Venn diagram for Mutually Exclusive Sets
4.4.
Collectively Exhaustive Sets
When union of all subsets
,….
comprise the whole sample space, ‘ ’, then
are called collectively exhaustive. These are denoted as:
,
. The
visualization is provided in the figure below (Fig. 7). One important point to note that
intersection of any two sets need not be null set.
Fig. 7. Venn diagram for Collectively Exhaustive Sets
4.5.
Partition of a Set
Partition of a set, ‘ ’ is the collection of mutually exclusive and collective exhaustive subsets
. It is denoted as:
where,
The visualization is provided below, it is to be noted that all the subsets
for
.
are
non-empty, non-overlapping events and the covering whole of set ‘ ’.
.
Fig. 8. Venn diagram for Set Partition
4.6.
Complements
Complements of a set ‘ ’ is the set of events in ‘ ’ that do not belong to ‘ ’. Complements of
any event ‘ ’ are denoted as:
. The visualization is provided in the fig. 9. Any
event or subset, other than those within the boundary of ‘ ’, are complements of ‘ ’, denoted
as A in the figure.
Fig. 9. Venn diagram for Complement
4.7.
De Morgan’s Law
When in a set identity, all the subsets are replaced by their complements, all unions by
intersections, and all intersections by unions, the identity is preserved. For example, if ‘ ’
and ‘ ’ are subsets of a set ‘ ’, then
,
….…………………………..(1)
Here, AB is shortened form of A  B . The visualization for this example is provided in the
following figure (Fig. 10).
S
Fig. 10. Example of De Morgan’s Law
De Morgan’s Law for set operation can be expanded or demonstrated using three subsets,
‘ ’, ‘ ’ and ‘ ’, all belong to ‘ ’. With repeated application of the law, it can be denoted as:
………………………………..(2)
Using the identity,
, one can find for three subsets:
………………………………..(3)
And, using the other identity,
, one can get:
……………………………….(4)
Similarly,
………………………….(5)
Thus, applying De Morgan law on both sides of the Eqn. 2, one can get:
……………………………..(6)
This is exactly the same taking Eqn. 3 for LHS and Eqn. 5 for RHS. One can draw a similar
diagram for three subsets as shown here (Fig. 10) to understand the application of De
Morgan’s Law.
4.8.
Duality Principle
This is basically an extension of De Morgan’s Law. Since it is known that for a set, ‘ ’,
and
, for identity like, Eqn. 6, if all the unions are replaced by intersections,
all intersections by unions and the set ‘ ’ by
and the set
by , the identity is still
preserved.
Thus
leads to
.
leads to
4.9.
, similarly,
Symmetric Difference
For any two sets
and , the set of elements which belongs to only one on the sets (not both)
is called Symmetric Difference Set. The Venn diagram for visualization of this set operation
is given in Fig. 11 below.
Fig. 11. Symmetric Difference Set
4.10.
Cartesian Product
For any two set of events ‘ ’ and ‘ ’, a new set, that comprises all the possible ordered pair
of elements belong to ‘ ’ and ‘ ’ (
‘ ’. It is denoted as
) respectively, is called Cartesian product of ‘ ’ and
.
Here, an ordered pair is a collection of objects having two coordinates, such that one can
uniquely determine any object. Thus, first coordinate is ‘ ’ and the second coordinate is ‘ ’
then the ordered pair is (
4.11.
), which is not same as (
).
Power Set
The Power Set of any set, ‘ ’ is the set whose elements are all possible subsets of ‘ ’ itself.
e.g. Power Set of
. is
5. Special Sets
A set ‘ ’ is called open set if any point ‘ ’ in ‘ ’ always belongs to the set, even though it
moves to some particular direction. The complement of an open set is called closed set.
Example of open set and closed set are given in Fig. 12. The points (
) satisfying the
condition
are coloured in blue. The points (
) satisfying the condition
are coloured in red. Any point within the area in red form an open set.
Whereas, the union of red and black points, forms closed set.
Fig. 12. Special Set [source: Wikipedia.org]
5.1.
Borel Set
Suppose
is an infinite sequence of sets in ‘ ’. If the union and intersection of
these sets also belongs to ‘ ’, then
5.2.
is called a Borel Field or Set.
Sigma-Algebra
A - algebra of a set ‘ ’ is a non-empty collection Ω of subsets of S (including S itself) that
is closed under complementation and countable unions of its members. e.g. if
one possible - algebra on
,
is:
6. Probability Space
Though Theory of Probability discussed already in previous lecture, here the relevance of it in
set operations are described in brief. According to Theory of Probability, or Ω is called as
certain event, its elements as experimental outcomes and its subsets as events. Here the empty set
element
is called as impossible event and the event
consists of single
only is called as an elementary event. It should be kept in mind that while
applying Theory of Probability to physical problems, the identification of experimental
outcomes is not always unique. A single performance of an experiment is called trial.
7. Concluding Remarks
Set Theory defines the various interrelationships among the elements or subsets of a set. The
set operations also classify various kinds of set properties of the elements. The details of
Axioms of Probability that deals with probabilistic properties of events and sets will be
discussed in the next lecture.