Set Theory: Maths Notes for NDA, CDS, AFCAT & CAPF


A set is a well-defined collection of objects, known as elements or members of the set. Sets are usually denoted by capital letters & elements are usually denoted by small letters. If ‘a’ is an element of a set A, then we write a ϵ A (a belongs to A) otherwise a ∉A (a doesn’t belong to A).

Representation of sets: 
1.Tabular or Roster form → In this form, elements are listed within the pair of brackets { } and are separated by commas.
e.g.— N = {1, 2, 3, 4 ….} is a set of natural numbers 
2.Set-builder or Rule form: In this form, set is describe by a property that its member must satisfy.
e.g. — A = {x : x is natural number less than 10}
3.Statement form: In this representation, well defined description of the elements of the set is given. 
e.g. — The set of all even numbers less than 10. 

Different Types of sets: 

1.Null set → A set which does not contain any element is called a null set or an empty set or a void set.

2.Singleton set → A set which contain only one element.

3.Finite set → A set is called a finite set, if it is either void or its elements can be counted 
The number of distinct elements of a finite set A is called the cardinal number & it is denoted by n(A).

4.Infinite set → A set which has unlimited number of elements is called infinite set.

5.Equivalence sets: Two finite sets A and B are equivalent, if their cardinal numbers are same.

6.Equal sets: Two sets are said to be equal if both have same elements. 
Note:– Equal sets are equivalent but equivalent sets may or may not be equal.

7.Subset: If every element of set A is an element of set B, then A is called a subset of B sit is denoted by A ⊆ B.

8.Superset → If set B contains all elements of set A, then B is called superset of A & it is denoted by B ⊇ A.

9.Proper subset → A set A is said to be a proper subset of set B, if A is a subset of B & A is not equal to B. It is denoted by A ⊂ B.

10.Universal set → Universal set is a set which contains all objects, including itself. It is denoted by U.

11.Power set → The set of all the possible subsets of A is called the power set & is denoted by P (A). 

Note:- 
1. The total number of subsets of a finite set containing n elements is 2ⁿ.
2. The total number of proper subsets of a finite set containing n elements is (2ⁿ –1).
3. If a set A has n elements, then its power set will contain 2ⁿ elements. 

Operations on sets: 
1. Union of two sets: The union of two sets A and B is the set of elements which are in A, in B or in both A & B. The union of A & B is denoted by A ∪ B. 
2. Intersection of two sets: The intersection of A & B is the set of all those elements which belong to both A & B & is denoted by A ∩ B. 
3. Disjoint of two sets: Two sets A & B are said to be disjoint if they don’t have any common element (i.e. A ∩ B = ϕ). 
4. Difference of two sets: The difference of sets A & B is the set of all those elements of A which do not belong to B. & is denoted by (A – B) or A\B.
5. Symmetric difference of two sets : The symmetric difference of sets A & B is the set (A – B) ∪ (B – A) and is denoted by A ∆ B. 
6. Complement of a set: The complement of a set A is the set of all those elements which are in universal set but not in A. It is denoted by A^1 or A^C or U – A. 

Laws of Algebra of sets: 
1. (a) A ⊆ A  ⋁  A
(b) ϕ ⊆ A ⋁ A
(c) A ⊆ U, ⋁ A in ∪
(d) A = B ⇔ A ⊆ B, B ⊆ A.

2. Idempotent laws:
(a) A ∪ A = A
(b) A ∩ A = A

3. Identity laws:
(a) A∪ ϕ = A 
(b) A∩ ϕ = ϕ  
(c) A ∩ U = A
(d) A ∪ U = U 

4. Commutative law
(a) A ∪B = B ∪A
(b)A ∩B = B ∩ A

5. Associative laws
(a) (A ∪B) ∪C = A ∪ (B ∪ C) 
(b) (A∩ (B ∩ C) = (A ∩ B) ∩ C

6. Distributive law
(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

7. De-Morgan’s law
(a) (A∪B)^' = A^' ∩ B^'
(b) (A∩B)^' = A^'∪B^'

8. (a) A – (B ∪C) = (A – B) ∩ (A – C) 
(b) A – (B ∩ C) = (A – B) ∪ (A – C) 
(c) A – B = A ∩ B^' = B^'– A^'
(d) A – (A – B) = A ∩ B
(e) A – B = B – A ⇔ A = B
(f) A ∪B = A ∩ B ⇔ A = B
(g) A ∪A^' = U
(h) A ∩ A^' = ϕ

Important results: 
1. n(A ∪ B)= n(A) + n(B) – n(A ∩ B)
2. n(A – B) = n(A) – n (A ∩ B) 
3. n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
4. n(A^' ∪ B^') = n ( A ∩ B)' = n (U) – n(A ∩ B) 
5. n(A^'∩ B^') = n(A ∪ B)' = n(U) – n(A ∪ B).
6. n (A ∆ B) = n (A) + n(B) – 2n(A ∩ B).
7. n(A') = n(U) – n(A). 
8. n(A ∩ B’) = n(A) – n(A ∩ B). 

Solved Examples

1. The number of proper subsets of the set {1, 2, 4, 6} is: 
Sol. Number of proper subsets of the set {1, 2, 4, 6} = 2⁴– 1 = 16 – 1 = 15.


3. In a group of 500 students, there are 475 students who can speak Hindi and 200 can speak English. What is the number of students who can speak Hindi only ? 

Sol. n(H) = 475, n(E) = 200, n (H ∪ E) = 500
n(H ∩ E) = n (H) + n(E) – n(H∪ E)
= 475 + 200 – 500 = 175. 
Number of students who can speak Hindi only 
= n(H) – n(H∩ E)
= 475 – 175 = 300 

4. Let n(U) = 800, n(A) = 250, n(B) = 300, n(A∩B) = 350 then n(A' ∩ B') is equal to 

Sol. n(A ∪ B) = n(A) + n(B) – n(A ∩ B) 
= 250 + 300 – 350 
= 200
n(A' ∩ B') = n(A ∪ B)' = n(U) – n(A ∪B)
= 800 – 200 
= 600

5. Let U = {x ϵ N : 1 ≤ x ≤ 8} be the universal set, N being the set of natural numbers. If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}. Then what is the complement of (A – B) ? 

Sol. A = {1, 2, 3, 4) B = {2, 4, 6, 8)
(A – B) = {1, 3} 
(A – B)' = U – (A – B) 
= {1, 2, 3, 4, 5, 6, 7, 8} – {1, 3} 
= {2, 4, 5, 6, 7, 8}

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