**Set Theory**

**Sets**

**set**is a collection of well-defined distinct objects.

**Collection of brilliant students of your class.**

**Collection of three months of a year.**

**Notation of a Set**

** Representation of a Set**

### 1. **Description Method**

### 2. **Roster Method or Tabular Form **

### 3. **Rule Method or Set Builder Form **

*is a factor of 36}.*

**Cardinal Number
of a Set**

** Types of Sets**

**Finite Set **

### **Infinite Set **

**infinite set**.

** Empty Set **

### **Singleton Set **

**Equal Sets **

**Equivalent Sets **

**if they have the same number of elements, the elements may or may not be the same. Thus, two sets A and B are equivalent if**

*n*(A) =

*n*(B).

*x*:

*x*is a prime number less than 19}, then

*n*(A) = 7 =

*n*(B).

**Overlapping Sets **

**if they have at least one element in common.**

**Disjoint Sets **

**if they have no element in common.**

*x*:

*x*is a student of ABC school} and set B = {

*x*:

*x*is a student of XYZ school} are disjoint sets as no student can study in both the schools at any one point.

** Complement of a Set**

**A is the set of all elements in the universal set which are not in set A. It is denoted by A’ and is read as ‘complement of A’.**

**Solved Examples:**

**Example 1: **Express each of the following sets in the form as
required.

a. The set of
integers between –3 and 3 (Roster method)

b. C = {*x *:
*x *is a letter in the word LOLLIPOP} (Description method)

**Solution: **

a. {–2, –1, 0,
1, 2}

b. C = {letters of the word LOLLIPOP}

**Example 2: **Write the following sets in set builder form.

a. The set A
of even natural numbers lying between 5 and 20.

b. B = {10,
20, 30, 40, …}

**Solution: **

a. A = {*x *| *x *=
2*k*, 3 ≤ *k *≤ 9, *k *ϵ** N**}

b. B = {*x *| *x *=
10*n*, *n *ϵ** N**}

**Example 3: **Write the following sets in roster form.

P = {*x *: *x *=
*n*^{2} + 2, *n *ϵ** N **and *n *≤ 5}

**Solution: **

P = {3, 6, 11, 18, 27}

**Example 4: **Find the cardinal number of the following sets.

a. A = {*x
*: *x
*= *n*^{2} + 1, *n
*ϵ** N **and *n *≤ 4}

b. B = {*x
*: *x
*is a day
of a week}

c. D = {*x
*: *x
*is a
letter of the word ‘NATIONAL’}

**Solution:
**

a. Here,
A = {2, 5, 10, 17} which has 4 elements.

Hence, *n
*(A) = 4

b. Here,
B = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

Hence,
*n
*(B) = 7

c. Here,
D = {N, A, T, I, O, L}

Hence,
*n
*(D) = 6

**Example
5: **State
whether each of the following statements is true or false when

A =
{letters of the word ‘NUMERAL’} and B = {letters of the word ‘MATERIAL’}.

a. A and B are equal sets.

b. A and B are equivalent sets.

c. A and
B are disjoint sets.

**Solution:
**

Here, A
= {N, U, M, E, R, A, L} and B = {M, A, T, E, R, I, L}

a. False as the elements are not identical

b. True as *n *(A) = *n
*(B) = 7

c. False as elements M, E, R, L and A are in both the sets.

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