Table of contents  
What is a Set?  
Types of Sets  
Representation of Sets  
Operations on Sets  
Laws of Sets 
A set is a collection of welldefined individual objects that form a group. A set can contain any group of items, such as a set of numbers, a day of the week, or a vehicle. Each element of the set is called an element of the set. Curly braces are used to create sets.
A very simple example of a set is: Set A = {1,2,3,4,5}. There are various notations for representing the elements of a set.
Do you Know?
The number of elements in a set is called its cardinal number and is written as n(A). A set with cardinal number 0 is called a null set while that with cardinal number ∞ is called an infinite set.
Union of the sets A and B, denoted by A ∪ B, is the set of distinct elements that belong to set A or set B, or both.
n(A U B) = n(A) + n(B) – n(A ∩ B) 
Here,
n(A U B) = Total number of elements in A U B; is called the cardinality of a set A U B
n(A) = Number of elements in A; is called the cardinality of set A
n(B) = Number of elements in B; is called the cardinality of set B.
n(C) = Number of elements in C; is called the cardinality of set C.
If A, B and C are 3 finite sets in U then,
n(A∪B∪C)= n(A) +n(B) + n(C)  n(B⋂C)  n (A⋂ B) n (A⋂C) + n(A⋂B⋂C) 
Example: Let U be a universal set consisting of all the natural numbers until 20 and set A and B be a subset of U defined as A = {2, 5, 9, 15, 19} and B = {8, 9, 10, 13, 15, 17}. Find A ∪ B.
Solution : We know, n(A U B) = n(A) + n(B) – n(A ∩ B)
Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
A = {2, 5, 9, 15, 19}
B = {8, 9, 10, 13, 15, 17}
A ∪ B = {2, 5, 8, 9, 10, 13, 15, 17, 19}
Example: If A = { 3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, and C = {11, 13, 15}, then find B ∩ C and A ∩ B ∩ C.
Ans: Given,
A = { 3, 5, 7, 9, 11} , B = {7, 9, 11, 13} , C = {11, 13, 15}
B ∩ C = {11, 13}
A ∩ B ∩ C = {11}
The difference between sets is denoted by ‘A – B’, which is the set containing elements that are in A but not in B. i.e., all elements of A except the element of B.
Some of the properties related to difference of sets are listed below:
Example : If X = {11, 12, 13, 14, 15}, Y = {10, 12, 14, 16, 18} and Z = {7, 9, 11, 14, 18, 20}, then find the following:
(i) X – Y – Z
(ii) Y – X – Z
(iii) Z – X – Y
Ans: Given,
X = {11, 12, 13, 14, 15}
Y = {10, 12, 14, 16, 18}
Z = {7, 9, 11, 14, 18, 20}
(i) X – Y – Z = {11, 12, 13, 14, 15} – {10, 12, 14, 16, 18} – {7, 9, 11, 14, 18, 20}
= {13, 15}
(ii) Y – X – Z = {10, 12, 14, 16, 18} – {11, 12, 13, 14, 15} – {7, 9, 11, 14, 18, 20}
= {10, 16}
(iii) Z – X – Y = {7, 9, 11, 14, 18, 20} – {11, 12, 13, 14, 15} – {10, 12, 14, 16, 18}
= {7, 9, 20}
The complement of a set A, denoted by A^{C }is the set of all the elements except the elements in A. Complement of the set A is U – A.
Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8}. Find A^{C}
Ans: A^{C} = U  A = {1, 3, 5, 7, 9, 10}
We can represent it in setbuilder form, such as:
A × B = {(a, b) : a ∈ A and b ∈ B}
Example: set A = {1,2,3} and set B = {Bat, Ball}, then;
A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}
(1) De Morgan’s Laws: For any two finite sets A and B;
(i) A – (B U C) = (A – B) ∩ (A – C)
(ii) A  (B ∩ C) = (A – B) U (A – C)
De Morgan’s Laws can also we written as:
(i) (A U B)’ = A' ∩ B'
(ii) (A ∩ B)’ = A' U B'
(2) Complement Laws: The union of a set A and its complement A’ gives the universal set U of which, A and A’ are a subset.
A ∪ A’ = U
Also, the intersection of a set A and its complement A’ gives the empty set ∅.
A ∩ A’ = ∅
For Example: If U = {1 , 2 , 3 , 4 , 5 } and A = {1 , 2 , 3 } then A’ = {4 , 5}. From this it can be seen that
A ∪ A’ = U = { 1 , 2 , 3 , 4 , 5}
Also
A ∩ A’ = ∅
(3) Law of Double Complementation: According to this law if we take the complement of the complemented set A’ then, we get the set A itself.
(A’)’ = A
In the previous example we can see that, if U = {1 , 2 , 3 , 4 , 5} and A = {1 , 2 ,3} then A’ ={4 , 5}. Now if we take the complement of set ‘A’ we get,
(A’)’ = {1 , 2 , 3} = A
This gives back the set A itself.
(4) Law of empty set and universal set: According to this law the complement of the universal set gives us the empty set and viceversa i.e.,
∅’ = U And U’ = ∅. This law is selfexplanatory.
Example: If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).
Ans: A ∩ B = {a, b, c} ∩ {c, d, e, f}
A ∩ B = { c }
=A ∩ C = { a, b, c } ∩ { c, d, e }
A ∩ C = { c }
Then (A ∩ B) ∪ (A ∩ C) = { c }
Example: Give examples of finite sets.
Ans: The examples of finite sets are: Set of months in a year
 Set of days in a week
 The Set of natural numbers less than 20
 Set of integers greater than 2 and less than 3
Example: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then find (A – B)′.
Ans: A – B is a set of member which belong to A but do not belong to B
∴ A – B = {3, 5, 7, 9, 11} – {7, 8, 9, 10, 11}
A – B = {3, 5}
According to formula, (A − B)′ = U – (A – B)
∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} – {3, 5}
So, (A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.
EduRev Tip: Any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅ , is also a subset of any given set X. The empty set is always a proper subset, except of itself. Every other set is then a subset of the universal set.
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1. What are the different types of sets in set theory? 
2. How are sets represented in set theory? 
3. What are some common operations performed on sets in set theory? 
4. What are the laws of sets in set theory? 
5. What are some important formulae in set theory for CAT exam preparation? 

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