Sets Definition and Examples5 min read

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 :

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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 : 

  1. 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}
  2. 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}
  3. 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 :

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 An = { x | x is an integer and x ≤ n}. Then {An | n is an integer} is a family of sets.

Singleton Set :

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

example :

  1. {0} is a singleton set whose only element is 0.
  2. {7} is a singleton set whose only element is 7.
  3. {-115} is a singleton set whose only element is -115.

Intervals in R : 

For any real numbers a and b, we define the intervals as the sets given below:
  1. (a, b) = { x | x ∈ R and a < x < b}
  2. (a, b] = { x | x ∈ R and a < x ≤ b}
  3. [a, b) = { x | x ∈ R and a ≤ x < b}
  4. [a, b] = { x | x ∈ R and a ≤ x ≤ b}

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