Set Theory

Basic Terminology

a) Set:
    

A set is a collection of well-defined objects. Each individual object is called an element of that set.
For example- the days of the week form a set as
D = { Monday, Tuesday, Wednesday, Thursday, Friday, 
         Saturday, Sunday }                                                         
Tuesday is an element of the set D. We write it as
        Tuesday   D
However, we are interested in using sets for probabilities.

b) Experiment:
    

An experiment is defined as any sort of operation whose outcome cannot be predicted in advance with certainty, the sample space S for such an experiment is the set of all possible outcomes that might be observed. For example, rolling a six-sided dice is an experiment.

c) Event:
     

An event is defined as a subset of the sample space, which contains any element of a sample space.

d) Sample Space:
    

A sample space is the universal set pertinent to a given experiment. The sample space is the set of all possible outcomes of an experiment.

e) Null Set:
    

It is a set containing no elements.

f) Universal Set:
    

It is the set containing everything in a given context.

g) Mutually Exclusive Sets or Disjoint Sets:
    

If there are no elements common to both sets under consideration, they are known as mutually exclusive sets or disjoint sets.
For example, if A = { 1, 2 }, B = { 3, 4 }, then A  B = 

NOTATIONS FOR A SET:

A set can be written in either form:

a) Roster Notation: To list all the elements of the set one by one.
eg., A = { 1, 2, 3, 4, 5, 6 }

b) Builder Notation: It gives a rule to follow that will tell us how to build the roster.
eg., A = { x  0 < x < 6, x is a whole number }

OPERATIONS:

a) Union of sets:

The union of two sets, A and B, written as A U B, is the set that consists of all the elements that belong to A or B or both. For example- A = { 1, 2, 3, 5 } and B = { 1, 2, 4, 6, 7 },
then A U B = { 1, 2, 3, 4, 5, 6, 7 }.

b) Intersection of sets:

The intersection of two sets, A and B, written as A B is the set that consists of all elements that belong to both A and B.
So for the above example, A B = { 1, 2 }.

c) Complement of a set:

 Complement of a given set is the set containing all the elements in the universal set that are not the members of the given set.