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.