**Set :**

A given collection of objects is said to be well defined, if we can definitely say whether a given particular object belongs to the collection or not. A well-defined collection of objects is called a set.

**example :**

- The collection of all vowels in the English alphabet contains five elements, namely a, e, i, o, u. So, this collection is well defined and therefore, it is a set.
- The collection of all odd natural numbers less than 10 contains the numbers 1, 3, 5, 7, 9. So, this collection is well defined and therefore, it is a set.
- The collection of all prime numbers less than 20 contains the numbers 2, 3, 5, 7, 11, 13, 17, 19. So, this collection is well defined and therefore, it is a set.
- The collection of all rivers of India, is clearly well defined and therefore, it is a set. Clearly, river Ganga belongs to this set while river Nile does not belong to it.
- The collection of most dangerous animals of the world is not a set, since no rule has been given for deciding whether a given animal is dangerous or not.
- The collection of five most renowned mathematicians of the world is not a set, since there is no criterion for deciding whether a mathematician is renowned or not.

**Element :**

Let X be any set. The objects belonging to X are called elements of X, or members of X. If x is an element X, then we say that x belongs to X and denote this by x ∈ X. If x does not belong to X, then we write x ∉ X.

A set is represented by listing all its elements between the brackets { } and by separating them from each other by commas, if there are more than one element.

**example :**

- The set of all natural numbers (i.e., the set of all positive integers) is denoted by N or Z
^{+}. That is, N = {1, 2, 3, 4, … }. - The set of all non-negative integers is denoted by W; that is W = {0, 1, 2, 3, …. }.
- Z denotes the set of all integers.
- Q denotes the set of all rational numbers.
- The set of all real numbers is denoted by R.
- The set of all positive real numbers is denoted by R
^{+}. - The set of all positive rational numbers is denoted by Q
^{+}. - C denotes the set of all complex numbers.

**There are two methods of describing a set.**

**1. Roster form or Tabulation Method :**

Under this method, we list all the members of the set within braces { } and separate them by commas.

**example : **

- A = set of all factors of 24

All factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

A = {1, 2, 3, 4, 6, 8, 12, 24} - B = set of all prime numbers between 50 and 70

All prime numbers between 50 and 70 are 53, 59, 61, 67

B = {53, 59, 61, 67} - F = set of all months having 30 days

We know that the months having 30 days are April, June, September, November

F = {April, June, September, November}

**2. Set builder form representation :**

A set may be represented with the help of certain property or properties possessed by all the elements of that set. Such a property is a statement which is either true or false. Any object which does not possess this property will not be an element of that set.

In order to represent a set by this method we write between the brackets { } a variable x which stands for each element of the set. Then we write the property (or properties) possessed by each element x of the set. We denote this property by p(x) and seperate x and p(x) by a symbol: or |, read as “such that”.

Thus, we write **{ x | p(x)} or { x : p(x)} **to represent the set of all objects x such that the statement p(x) is true. This representation of a set is called “set builder form” representation.

**example :**

(1) Let P be the collection of all prime numbers. Then it can be represented in the set builder form as

**P = {x|x is a prime number}**

(2) Let X be the set of all even positive integers which are less than 15. Then

**X ={ x | x is an even integer and 0 < x < 15 }**

**X ={ 2, 4, 6, 8, 10, 12, 14 }**

Empty Set :

Empty Set :

The set having no elements belonging to it is called the empty set or null set and

is denoted by the symbol Φ .

**example :**

Let **X = {x|x is an integer and 0 < x < 1} ** Then X is a set and there are no elements in X, since there is no integer x such that 0 < x < 1. Therefore, X is the empty set.

**Equal Sets :**

Two sets A and B are defined to be equal if they contain the same elements, in the sense that, x∈A ⇔ x∈B

**example :**

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

**Finite and Infinite Sets :**

A set having a definite number of elements is called a finite set. A set which is not finite is called an infinite set.

**example :**

- The set Z
^{+}of positive integers is an infinite set. - {a, b, c, d} is a finite set, since it has exactly four elements.

**Family of Sets :**

A set whose members are sets is called a family of sets or class of sets. Note that a family of sets is also a set.

**example :**

For any integer n, let A

_{n}= { x | x is an integer and x ≤ n}. Then {A

_{n}| n is an integer} is a family of sets.

**Singleton Set :**

A set containing exactly one element is called a singleton set.

**example :**

- {0} is a singleton set whose only element is 0.
- {7} is a singleton set whose only element is 7.
- {-115} is a singleton set whose only element is -115.

**Intervals in R : **

- (a, b) = { x | x ∈ R and a < x < b}
- (a, b] = { x | x ∈ R and a < x ≤ b}
- [a, b) = { x | x ∈ R and a ≤ x < b}
- [a, b] = { x | x ∈ R and a ≤ x ≤ b}