Sets and Their Representation

Imagine a world where no categories are differentiated in order to memorize and classify things independently; such a world would be chaotic and disorganized. A set is a group or collection of well-defined data represented by capital letters of the English alphabet which can be represented in set-builder form or roster form, they are represented in curly braces {} on both sides which were developed to describe the collection of data.

Set

A collection of data with defined limits is referred to as a “Set.” In mathematics, a set is a tool that is used to classify and collect data that belongs to the same category, even if the components in the set are all distinct, they are all similar since they belong to the same group. A set is surrounded by curly brackets on both sides.

The number of elements that are included in the set acts as a factor that determines the order of the set. It is possible to denote the total number of elements in a set using the notation n(A), where A stands for the set. When dealing with infinite sets, the order of the set can never be exhausted.

For example, 

1. Consider a set of distinct indoor games A,

A= {Musical Chairs, Charades, Ludo, Carrom, Chess}

Although the games are different, they all belong to the same group which is indoor games. Here number of elements in A

n(A)= 5

2. Consider a set X of letters of the word ARITHMETIC,

X={A,R,T,I,H,M,E,C}

Here we will not repeat I,T as they have already been listed in the set. Here number of elements in X

n(X)= 8

Conventions of a Set

We denote the number sets by following notations:

• N is a representation of a collection of Natural Numbers
• Z is a representation of an Integer set
• W stands for a collection of Whole Numbers
• R is the abbreviation for a collection of real numbers
• Q is a logical number collection
• T stands for a collection of irrational numbers
• Z+ represents a set of positive integers
• Z– is a symbol for a collection of negative integers
• Q+ represents a collection of Positive Rational Numbers
• Q– is the abbreviation for a collection of negative rational numbers
• R+ denotes a collection of Real Positive Numbers
• R– is a notation for a collection of Negative Real Numbers

Set- Representation

Both the Roster form and the Set-Builder form are ways that sets can be represented. The only difference between these two forms is how the information is shown. Both forms can be used to say the same thing. One of these forms is known as the Set-Builder form, and the other is known as the Roster Form. A set P of the first eight prime numbers can be represented by,

P= {2,3,5,7,11,13,17,19}
When it comes to representing sets, there are two distinct approaches:

1. Roster or Tabular form

In Roster Form, which is also called Tabular form the elements are in Curly brackets. All of the parts are listed inside, with commas between each one. It is irrelevant in which sequence elements are listed, however, elements cannot be repeated. The easiest approach to represent data in groups is in roster format.

For example, the set for the table of 3 will be, A= {3, 6,9,12,15,18,21,…}

The properties of this type of set are-

• The arrangement in the Roster form does not necessarily have to be in the same order every time. For example, A= {a, e, i, o, u} is equal to A= {e, i, a, u, o}
• When represented in Roster form, the set does not contain an instance of each element occurring more than once. for instance, the word “maths” will be written as A= {a, m, t, s, ,h}
• The elements of finite sets can be represented in one of two ways: either with all of the elements themselves or as dots in the centre of the representation if there are too many elements. Dots are used to denote the infinite sets at the conclusion of the sentence

2. Set-Builder Form

In Set-builder form, elements are displayed or represented by statements expressing their relationships. A=”a: statement” is the usual representation for Set-builder.

For example, A = {x: x = a², a ∈ N, a < 0}

The properties of this type of set are-

• The data should be organized according to a specific pattern in order for the set to be written in Set-builder form
• In the Set-builder form, you are required to use colons (:)
• The statement should be written after the colon in this case

Symbols Used in Set Builder Notation-

• ∈ means “is an element of”
• ∉ means “is not an element of”

Conclusion

Any well-defined collection of mathematical objects can form a set which we have learnt in the above article. It is possible to represent it using two different forms. They are set builder form and roster form. The roster form is also known as the tabular form. The number of elements present in the set is also known as its order. Depending on the total number of items contained within the set, the order of the set might either be finite or infinite.