BASIC CONCEPTS OF SETS

We often deal with a group or a collection of objects, such as a collection of books, a group of students, a list of states in a country, a collection of coins, etc. Set may be considered as a mathematical way of representing a collection or a group of objects.

Key Concept  - Set 

A set is a collection of well defined objects. The objects of a set are called elements or members of the set.

The main property of a set in mathematics is that it is well defined. This means that given any object, it must be clear whether that object is a member (element) of the set or not.

The objects of a set are all distinct, i.e., no two objects are the same.

Which of the following collections are well defined?

(1) The collection of male students in your class.

(2) The collection of numbers 2, 4, 6, 10 and 12.

(3) The collection of provinces in United States of America.

(4) The collection of all good movies.

(1), (2) and (3) are well defined and therefore they are sets. (4) is not well defined because the word good is not defined. Therefore, (4) is not a set.

Generally, sets are named with the capital letters A, B, C, etc. The elements of a set are denoted by the small letters a, b, c, etc.

Reading notation : 

If x is an element of the set A, we write x ∈ A.

If x is not an element of the set A, we write x ∉ A.

For example,

Consider the set A  =  {1, 3, 5, 9}

1 is an element of A, written as 1 ∈ A

3 is an element of A, written as 3 ∈ A

8 is not an element of A, written as 8 ∉ A.

Representation of a Set

A set can be represented in any one of the following three ways or forms.

(i)  Descriptive form

(ii)  Set-builder form or Rule form

(iii)  Roster form or Tabular form

Let us discuss the above different forms representation of a set in detail. 

Descriptive Form

One way to specify a set is to give a verbal description of its elements.

This is known as the descriptive form of specification.

The description must allow a concise determination of which elements belong to the set and which elements do not.

For example, 

(i) The set of all natural numbers.

(ii) The set of all prime numbers less than 100.

(iii) The set of all letters in English alphabets.

Set-Builder Form or Rule Form

Set-builder notation is a notation for describing a set by indicating the properties that its members must satisfy.

Reading notation :

A  =  {x : x is a letter in the word "dictionary"}

We read it as 

“A is the set of all x such that x is a letter in the word dictionary”

For example,

(i)  N  =  {x : x is a natural number}

(ii)  P  =  {x : x is a prime number less than 100}

(iii)  A  =  {x : x is a letter in the English alphabet}

Roster Form or Tabular Form

Listing the elements of a set inside a pair of braces {  } is called the roster form.

For example,

(i)  Let A be the set of even natural numbers less than 11. 

In roster form we write A  =  {2, 4, 6, 8, 10}

(ii) A  =  {x : x is an integer and  -1 ≤ x < 5}

In roster form we write A =  {-1, 0,1, 2, 3, 4}

Representation of Sets in Different Forms

Important Points

(i)  In roster form each element of the set must be listed exactly once. By convention, the elements in a set should not be repeated.

(ii)  Let A be the set of letters in the word “follow”,

That is,  A  =  {f, o, l, w}. So, in roster form of the set A the following are invalid.

{f, o, w} -------> (not all elements are listed)

{f, o, l, l, w} -------> (element ‘f’ is listed twice)

(iii)  In a roster form the elements in a set can be written in any order.

The following are valid roster form of the set containing the elements 2, 3 and 4.

{a, b, c}

{a, c, b}

{c, b, a}

Each of them represents the same set.

(iv)  If there are either infinitely many elements or a large finite number of elements, then three consecutive dots called ellipsis are used to indicate that the pattern of the listed elements continues, as in  

{1, 3, 5, 7,......}  or  {2, 4, 6, 8, 10,........60}

(v) Ellipsis can be used only if enough information has been given so that one can figure out the entire pattern.

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