Sets and their Representation

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.