**1. Various types of sets:**

**Finite set**

A set which contains limited number of elements is called a finite set.

**Example1. A = ****{1, 3, 5, 7, 9}. **

Here A is a set of five positive odd numbers less than 10. Since the number of elements is limited, A is a finite set.

2. A grade 5 class is a finite set, as the number of students is a fixed number.

**Infinite set**

A set which contains unlimited number of elements is called an infinite set.

1. The set of natural numbers N, is an infinite set as the counting of numbers does not come to an end.

2. The set of integers is an infinite set.

**Singleton set**

A set which contains only one element is a singleton set.

**Example 1: **

A {set of even prime numbers}

Now A = {2}.

The only even prime number is 2. All other prime numbers are odd.

Therefore A can contain only one element, namely 2.

Therefore A is a singleton set.

**Example 2: **

X = {x: x is neither positive nor negative}

Now, X = {0}, because it’s only 0 which is neither positive nor negative.

Therefore, X is a singleton set.

**Null set**

A Set which does not contain any element is called empty set or null set.

S = {x: x ∈Z, x = 1/n, n ∈ N}

N is natural number and Z is integer.

Since n is an integer, 1/n cannot be an integer. Therefore, S cannot contain an element x which is an integer.

**Note: **

1. The Empty set is denoted as { } or by the greek letter Φ

2. {{}} or {Φ} are not empty sets, because each contain one element, namely the empty set Φ itself.

**Cardinal Number of a set or Cardinality of a set:**

The cardinality of a set is the number of elements a set contains. It is denoted as n (A).

n (A) is read as the number of elements in set A

**Example 1: **

A = {1, 2, 3, 4, 5}

The cardinality of set A is 5.

It is denoted as n (A) = 5

** Note: **

**Example 1. **

Let X = { }, then n (X) = 0

Let Z = {{}} or Z = {Φ}, then n (Y) = 1

Example 2:

Cardinality of infinite set is not defined.

**Equivalent sets**

Two sets which have the same number of elements, i.e. same cardinality are equivalent sets.

P = {p. q. r, s, t} and Q = {a, e, i, o, u}

Since the two sets P and Q contain the same number of elements 5, therefore they are equivalent sets.

**Equal sets**

Two sets that contain the same elements are called equal sets.

A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {

**Overlapping sets**

Two sets that have at least one common element are called overlapping sets.

**Example 1: **

X = {1, 2, 3} and Y = {3, 4, 5}

The two sets X and Y have an element 3 in common. Therefore they are called overlapping sets.

**Example 2:**

A = {x: x is an even prime number}

B = {x: x**∈****2n,n****∈****N** }

The two sets A and B are overlapping sets because 2 is a common element in A and B.

**Disjoint sets**

C = {2, 4, 6} and D = {1, 3, 5}

The two sets C and D are disjoint sets as they do not have even one element in common.

**Example 1:**

E = {set of odd numbers} and F = {set of even numbers}

The two sets E and F are disjoint as no odd number is an even number nor any of even numbers is odd.

**Subset **

Set A is a subset of set B if every element of A is an element of set B.

If set A is a subset of set B, then it is denoted as A**C**B

**Example 1:**

Let A = {1, 2, 3} and B = {2, 3, 4, 1}

Since every element of set A is present in set B too, we say A is a subset of B.

**Example 2: **

Let A = {multiples of 4} B = {multiples of 2}

Now, X = {1, 4, 8, 12...} and Y = {1, 2, 4, 6, 8, 10, 12...}

Since every element of set X is also an element of set Y, therefore

X is a subset of Y.

**Note: **

**1. If two sets A and B are equal sets, then each one is a subset of the other. **

If A = {a, e, i, o, u} and B = {vowels of English alphabets}, then A = B.

But, note that A **C** B and B**C**A.

Therefore, if A **C** B and B**C**A, then A = B

**2. Every set is a subset of itself. **

A**C**A

3. empty set is a subset of every set.