**What is a Set?**

Sets represent a group of distinct objects. In mathematics, sets are a well-organised collection of objects represented in a set builder form or roasted form. Generally, we represent sets using curly braces {}.

The components of a set are called elements. Using the example given below, let’s understand the concept of sets and their representation.

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

Since a set is always represented by a capital letter, thus in the above equation, A is the set, and 1,2,3,4,5 are the elements of that set or the associate of that set. The elements in a set can be done in any order but never be repeated. “∈” is a Greek symbol used to represent the expression “belong to.”

A={a,e, i,o,u}, where we can say a ∈ A and b ∉ A, “a” belongs to A and “b” doesn’t belong to set A; therefore, it means a is the element of set A and on their other b is not an element of set A. But the elements are never repeated.

For example, we can write MATHEMATICS as

X={M, A, T, H, E, I, C, S} and will not repeat M, A, T as they are already on the set previously.

Set also has several notations which are,

N: the set of numbers

W: the set of whole numbers

Z: Set of integers

Q: Set of Rational numbers

T: Set of irrational numbers

R: Set of real numbers

Z+: Set of positive integers

Q+: Set of positive rational numbers

R+: Set of positive real numbers

**Order of Sets**

The order of a set refers to the total number of elements a set has. It defines the size of the text. Cardinality is another name for the order of a set.

A finite or infinite is a set of infinite order and finite order.

**Define representation of sets**

There are three forms of** representation of sets,** namely statement form, set builder form, and roster form. Now that you have understood the concept of sets let’s move on to understanding the different methods of **representation of sets.**

**Statement Form**

The statement form refers to the method of** representation of sets** where a well defined description for a set member is given. This well defined description of a set member is enclosed in curly braces. To give you a better understanding of the statement form, here is an example of a set of odd numbers less than 10.

You can rewrite this in statement form as

{Odd numbers less than 10}

**Set Builder Form**

The general form of set builder **representation of sets** is given below. Suppose we have a set A that encapsulates odd numbers from 1 to 10, and all the members are a multiple of 3.

Let’s consider A= {3,6,9}

As all the members are a multiple of 3. It can also be solved as

3×1=3

3×2=6

3×3=9

This set, when represented in set-builder form, can be rewritten as A = {x: x=3n, n ∈ N and 1 ≤ n ≤ 3}

There is a concept of Venn diagrams which are the best method to visualise the **representation of sets** in a simpler manner.

**Roster form**

The roster form is a listed manner of **representation of sets.** Let’s understand the concept of roster form with an example of a set of natural numbers less than 10.

Natural numbers are all the naturally occurring numbers excluding zero, invented by a renowned Indian mathematician Aryabhatta.

All the natural numbers less than 10 are 1,2,3,4,5,6,7,8,9.

Rosster form representation of the set is

N= {1,2,3,4,5,6,7,8,9}

**Types of Sets**

Now that you have become familiar with the different methods to represent a set. Let’s move on to understand the different types of sets.

There are six types of sets in mathematics, namely.

- Empty set
- Singleton set
- Finite set
- Infinite set
- Equivalent set
- Equal set
- Disjoint set

**Empty Set**

An empty set refers to a set that does not contain any element. The other names of empty sets are void set or null set. It is represented by {}.

**Singleton Set**

A singleton set is defined as a set with a single element.

**Finite set**

A set that contains a definite number of elements is termed a finite set.

**Infinite set**

An infinite set is the opposite of a finite set. In addition to that infinite set, it has an infinite number of elements.

**Equivalent set**

If there are two sets, A and B, and they have the same number of elements, they are equivalent sets. However, the order of elements does not matter.

**Equal set **

If there are two sets, A and B, they are said to be equal if they contain the same elements. However, the order of elements does not matter in equal sets.

**Disjoint set**

If there are two sets, A and B, and do not have any common element, we can say that A and B are disjoint sets.

**Conclusion**

The** set representation in the data structure** is an important concept if you are preparing for various competitive exams. In this** set representation in Data Structure pdf, **we have discussed the concept of sets and their representations. For example, Let A be the set of vowels in a collection of well-organised constituents, and as we can conclude or say, A is a well-organised set of constituents, and hence it is a set. In addition to that, we have discussed the various types of sets and subsets.