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

**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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