Sets, in mathematics, are an organized collection of objects and can be represented in setbuilder form or in roster form. In this section, students will get a detailed knowledge of the law of sets and some of the properties of the sets from class 11 onwards. The purpose of sets maths is to group a collection of related objects. It is important everywhere in mathematics because it helps for building the more complex mathematical structure.
Sets are represented as a collection of welldefined objects or elements and it does not change from person to person. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set.
Sets Theory
In, sets theory, we will learn about sets and it’s properties. It was developed to describe the collection of objects. You have already learned about the classification of sets here. The setÂ theory defines the different types of sets symbols and operation performed.
Operations on Sets
In set theory,Â the operations of the sets are carried when two or more sets combined to form a single set under some of the given conditions. The basic operations on sets are:
 Union of sets
 Intersection of sets
 A complement of a set
 Cartesian product of sets.
 Set difference
Sets Formulas
Some of the most important set formulas are:
For any three sets A, B and C 
n ( A âˆª B ) = n(A) + n(B) – n ( A âˆ© B) 
If A âˆ© B = âˆ…, then n ( A âˆª B ) – n(A) + n(B) 
n( A – B) + n( A âˆ© B ) – n(A) 
n( B – A) + n( A âˆ© B ) – n(A) 
n( A – B) + n ( A âˆ© B) + n( B – A) = n ( A âˆª B ) 
n ( A âˆª B âˆª C ) = n(A) + n(B) + n(C) – n ( A âˆ© B) – n ( B âˆ© C) – n ( C âˆ© A) + Â n ( A âˆ© B Â âˆ© C) 
Also, check the set symbols here.
Representation of Sets
The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either theÂ Statement form, Roster Form or Set Builder Form.
Statement Form
In statement form, the welldefined descriptions of a member of a set are written and enclosed in the curly brackets.
For example, the set of even numbers less than 15.
In statement form, it can be written as {even numbers less than 15}.
Roster Form
In Roster form, all the elements of a set are listed.
For example, the set of natural numbers less than 5.
Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….
Natural Number less than 5 = 1,2,3,4
Therefore the set is N = { 1, 2, 3, 4 }
Set Builder Form
The general form is, A = { x : property }
For example: Write the following sets in set builder form: A={2, 4, 6, 8}
Solution:
2 = 2 x 1
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
So, the set builder formÂ is A = {x: x=2n, n âˆˆ N and 1 Â â‰¤ n â‰¤ 4}
Also, Venn Diagrams are the simple and best way for visualized representation of sets.
Types of Sets
 Empty Set:Â A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ã˜.
 Singleton Set:Â A set which contains a single element is called singleton set
 Finite set:Â A set which consists of a definite number of elements is called finite set
 Infinite set:Â A set which is not finite is called infinite set
 Equivalent set: If the cardinal number of the two finite sets are equal, then it is called an equivalent set. I.e, n(A) = n(B)
 Equal sets:Â The two sets A and B are said to be equal if they have exactly the same elements
 Subsets: A set ‘A’ is said to be a subset of B if every element of A is also an element of B. Intervals are subsets of R
 Disjoint Sets: The two sets A and B are said to be disjoint if the set does not contain any common element
 Proper set: If A âŠ† B and A â‰ B, then A is called the proper set of B and it can be written as AâŠ‚B
What are the Elements of a Set?
Let us take an example:
A = {1, 2, 3, 4, 5 }
Since a set is usually represented by the capital letter. Â Here A is the set and 1, 2, 3, 4, 5 are the elements of the set or a member of a set. The elements that are written in the set are in any order and it cannot be repeated. All the set elements are represented in small letter in case of alphabets. Â Also, we can write it as 1 âˆˆ A, 2 âˆˆ A etc. The cardinal number of the set is 5. Some commonly used sets are as follows:
 N: Set of all natural numbers
 Z: Set of all integers
 Q: Set of all rational numbers
 R: Set of all real numbers
 Z+: Set of all positive integers
Complement of Sets
The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by Pâ€™.
Properties of Complement sets:
 P âˆª Pâ€² = U
 P âˆ© Pâ€² = Î¦
 Law of double complement : (Pâ€² )â€² = P
 Laws of empty/null set(Î¦) and universal set(U),Â Î¦â€² = U and Uâ€² = Î¦.
Properties and Laws
Commutative Property :

Associative Property :

Distributive Property :

Demorganâ€™s Law :

Complement Law :

Idempotent Law And Law of null and universal set :
For any finite set A

⇒ Learn more about De Morgan’s First Law here.
Example of Sets
Here are few sample examples, given to represent the elements of a set.
Example 1:
Write the given statement in three methods of representation of a set:
The set of all integers that lies between 1 and 5
Solution:
The methods of representations of sets are:
Statement Form: { I is the set of integers that lies between 1 and 5}
Roster Form: I = { 0,1, 2, 3,4 }
Setbuilder Form: I = { x: x âˆˆ I, 1 < x < 5 }
Example 2:Â
Find A U B and A n B and A – B.
If A = {a, b, c, d} and B = {c, d}.
Solution:Â
A = {a, b, c, d} and B = {c, d}
A U BÂ = {a, b, c, d}Â
A n B = {c, d} andÂ
A – B = {a, b}
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