Set theory was invented in the 18th century by a German Mathematician named George Cantor. His study focused on different operations involved in the sets. The term ‘sets’ helps to classify and summarise data appropriately. They play a very significant role in almost every field of mathematics, such as in probability, geometry, sequences, permutations, and combinations. Along with this, every branch of mathematics uses sets to define functions and relations. Since sets have a lot of applications in our everyday life, studying them is very essential. This article gives a detailed view of sets and their representations.

What are Sets?

A set is defined as the collection and classification of the given data. In mathematics, sets help to categorise the given data and arrange them accordingly. These sets may include anything that is universally accepted. For example, it may include the number system, attributes, statistical data, etc. Generally, a set is denoted by the respective capital letters.

Here are a few examples of sets:

• N: The set of all natural numbers (1 to ∞).
• S: The set of all square roots.
• Q: The set of all bright colours in a map.
• G: The set of girls in a classroom.
• M: The set of all motorbikes in a city.
• V: The set of vowels.

Important Notations

The following are the universally accepted set notations:

• N: The set of all the natural numbers.
• Z: The set of all the integers.
• Q: The set of all the rational numbers.
• R: The set of all the real numbers.
• Z+: The set of all the positive integers.
• Q+: The set of all the positive rational numbers.
• R+: The set of all the positive real numbers.

Representation of Sets

The representation of sets is flexible and can be done in different ways as per the requirements. There are three methods to represent a set: roster or tabular form, statement or descriptive form, and set-builder or rule form.

Roster or Tabular Form

In the roster method, the sets are represented using curly brackets { }. The data is enclosed inside the brackets and is separated by using a comma (,). For example, the set of all vowels is written in a roster form as  . It is called an extensional definition.

Here are some more examples of the sets represented in the tabular form:

• A set of all one-digit prime numbers can be represented as 
• The set of all positive integers less than 5 can be represented as 

Properties

The following are some of the properties of the roster form:

• While representing sets in roster form, the order of the set in which the elements are listed is irrelevant. It is not necessary to follow any kind of order. For example, the set of vowels can also be written as  .

• In the roster form, repetition of elements is not allowed. Here, every element should be unique in itself. For example, the set of all letters in the word ‘colour’ should be written as  .

Set Builder or Rule Form

In the set-builder form, the elements of a set are represented in a symbolic form. Here, all the elements in the set possess a common feature that is unavailable in the elements outside the sets. For example, X is the set of all two-digit natural numbers. It can be represented in a set-builder form such that, X = 

Here are some more examples of the set-builder method of representation:

• A =    
• B =   

Properties

The following are some of the properties of the rule form:

• The elements of the data should possess the same attributes.
• Using colon (:) is compulsory.
• The data of the elements should be systematically described after the colon.

Statement form

In the statement form, the elements of the set are provided in a well-defined format within the curly brackets. Hence, the statement form is also known as a descriptive form. For example, the set of all the one-digit prime numbers is written as 

Properties

The following are the properties of the statement or the descriptive form:

• A verbal description of the elements is essential.
• The description should be enclosed within the bracket.

Examples

1. Write the roots of the equation x2 + x – 2 = 0 in roster form.

Solution:

The first step is to solve the given equation x2 + x – 2 = 0.

It can be further simplified as (x-1) (x+2) = 0

Therefore, the roots of the equation are x = 1 and x = -2 respectively.

Hence, the roots of the equation can be written as   {1,-2}

2. Convert the set {x: x = a  Z+, a2 < 60} in roster form.

Solution:

In the given set, the elements are positive integers and perfect squares less than 60.

These numbers are 1, 4, 9, 16, 25, 36, and 49 and can be represented in roster form as  

{1,4,9,16,25,36,49}

3. Convert the set X =   in set builder form.

Solution:

The given set is a set of natural numbers that are perfect cubes.

Hence, it can be represented in set-builder form as

X = 

4. Convert the set X =   into set-builder form.

Solution:

In the given set of fractions, the denominators are natural numbers.

Hence, it can be represented in set-builder form as

X =  

5. Write the set A =    in the statement form.

Solution:

The given set A is the set of all the natural numbers.

Hence, it can be written in the statement form as 

Conclusion

In mathematics, sets are the collection of comprehensive and well-defined data. In simple words, the nature of the collected data possesses similar attributes.

The sets are denoted with the respective capital letter and are written in a curly bracket. These sets can be represented in two main forms called the roster form and the set builder form. In the roster method, the elements written are separated with commas, whereas in set-builder form, the elements are described using a colon. Both the methods are widely used in different branches of mathematics such as statistics, probability, and calculus.