Sets:


A well defined collection of distinct objects is called a set. Example: A = {a,e,i,o,u}

Important Rules for Writing a Set:

  • A set is always enclosed within opening and closing of the curly brackets i.e. '{' and '}'.
  • The members of a set are said to be the elements of the respective sets. Simply, objects that are kept in a set are said to be the elements of the set.
  • No elements can be repeated in one set. 

Representation of Sets:

There are three methods of representing a set. They are:

i) Listing method:

In this method, the elements of Sets are written in a listing order.
Example: A = {0,1,2,3,4}

ii) Description method:

In this method, the elements of Sets are written in a descriptive order,
Example: A = {whole numbers less than 5} 

iii) Set builder method:

In this method, the elements of Sets are written in mathematical formulae using notation.
Symbol '∈ ' is used to represent "belongs to".
Symbol ''∉' is used to represent "doesn't belong to".
Example: A = {x: x∈W; x<5} 

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Types of sets:

One set only!

Finite Sets:

Those sets having finite or countable number of elements are said to be finite sets. 
Example: A={a,b,c,d,e,f}
In the above example, set A has only six elements and there would be limited or countable number of elements therefore, it is said to be finite set.

Infinite Sets:

Those sets having infinite or uncountable number of elements are said to be infinite sets
Example: A= {1,2,3,4,5,.........}
In the above example, set A has endless number of elements. The elements start from 1 and continue and keep continuing. Since, we cannot count the total number of elements present in the set therefore, it is said to be infinite set.

Empty or Null Sets:

Those sets having no elements or zero elements are said to be empty or Null Sets.
Example: A ={}
In the above example, there is no any element in the set A therefore, it is said to be empty or Null set.


Unit or Singleton Sets:

Those sets having only one element are said to be Unit or Singleton sets.
Example: A={q}
In the above example, in set A, there is only one element i.e. 'q', therefore it is said to be unit or singleton set.

Two sets!

Equal Sets: 

Sets having all of their elements same are called equal sets. If Set A and set B are two sets, then the elements of Set A are exactly the same elements of Set B and their cardinal number is also same, then these types of sets are said to be equal sets.
Eg: A = {a,b,c,d} and B = {a,b,c,d}
In the above mentioned example; the elements of set A and the elements of Set B are exactly the same therefore, they are said to be equal sets.

Equivalent Sets:

Sets having their cardinal numbers same are called equivalent sets. When set A and set B are two sets and the number of elements (cardinal number) of set A is equal to the set B but their elements are not same, such sets A and B are said to be equivalent sets.
Eg: A = {1,2,3,4} and B = {a,b,c,d}
In the above example, the elements of set A and set B are not same but the number of elements in both sets are equal therefore, they are said to be equivalent sets.

Overlapping Set: 

Sets that have some of their elements common are called overlapping sets. If Set A and Set B are two sets, element 'x' is the element of both sets then, such set A and set B are said to be overlapping sets.
Eg: A = {1,2,3,4,5} and B = {4,5,6,7}
In the above example, two elements of both the sets i.e. 4 and 5 are common therefore, they are said to be overlapping sets.

Disjoint Set: 

Sets that have all of their elements different are called disjoint sets. If Set A and Set B are two sets, and none of their elements are common then, set A and set B are said to be disjoint sets.
Eg: A = {1,2,3} and B = {4,5,6}
In the above example, none of the elements of both the sets are common therefore, they are said to be disjoint sets.

Subsets:


For our study purpose, we define reference set as the original set from which we form subsets, here.

The sub-group of any set is said to be the subset of that particular reference set. Generally, to find the number of subsets that can be formed by a particular reference set, we use the formula:
2^n, where 'n' refers to the cardinal number of the reference set.

A set is said to be the proper subset if its cardinal number is less than the reference set.

A set is said to be the improper subset if its cardinal number is equal to the reference set.

A set that consists all the elements of its subsets is said to be the universal set. Or, a set from which many other subsets can be formed is said to be the universal set.

For example:
A = {1,2,3,4,5,6,7,8,9,10}
B = {2,4,6,8}
C = {1,2,3,4,5,6,7,8,9,10}
D = {9,11}

We have,
B 'is a proper subset of ' A
B 'is a proper subset of ' C 

But, 
C 'is an improper subset of ' A

And,
Universal Set, U = {1,2,3,4,5,6,7,8,9,10,11}


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Set Operations:

We can perform four different operations on one or more than one sets. They are:

i) Union of Sets:

The result (sum) that we obtain by adding the elements of two sets is said to be the union of sets.

ii) Intersection of Sets:

The overlapping elements of two or more than two sets is said to be the intersection of sets. 
Simply, the repetitive elements in all the given sets is said to be the intersection of given sets.

iii) Difference of two Sets:

The unique elements of one particular set only is said to be the difference of two sets.

iv) Complement of a set:

Elements of a set which do not belong to the given set but is the element of the universal set is called complement of a set. Simply, the difference of the Universal set and the given set is said to be the complement of a set.


Cardinal Numbers:

The total number of elements present in one set is said to be the cardinal number of that particular set. 
Example: A = {p,r,a,c,h,i,t}

In the above example, we have seven elements in the set A i.e. p, r,a,c,h,i,t. Because there are seven elements, the cardinal number of the set A is 7.
It is written as n(A) = 7

Cardinality Relations of Sets:


1. Cardinality Relations of union of two sets:



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