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Study About Set Representation and Sets Theory Symbols in Detail

Study About Set Representation and Sets Theory Symbols in Detail

Set Theory is the study of sets and their properties in mathematics. A set is a collection of items. The objects of a given set are its elements. The study of such sets and the links that exist between them is known as set theory. Set theory has shown to be a very helpful tool in defining some of mathematics' most difficult and important structures. As a result, it is a significant component of the numerical ability/quantitative aptitude syllabus in a variety of competitive tests.


On that note, let’s discuss Set Representation and Set Theory Symbols in detail for in-depth study.

Sets Representation

Sets can be represented in two ways:


       Roster Form or Tabular form

       Set Builder Notation

Roster Form

All of the elements in the set are listed in roster form, separated by commas and enclosed by curly braces { }.


In Roster form, for example, if the set covers all leap years between 1999 and 2017,  it would be expressed as


A ={2000, 2004, 2008, 2012, 2016,}


In the roster form, the order of elements in a set is irrelevant; the order might be ascending or descending.


Furthermore, frequency is neglected when representing the sets. For example, if X represents a set containing all of the letters in the word APPLE, the correct Roster form representation would be


X ={A, P, L, E }= {E, L, P, A} 


Set Builder Form

In set builder form, all elements share a property. This property does not apply to objects that are not members of a set.


For instance, if set S contains only even prime numbers, it is denoted as


S= { x: x is an odd natural number}


where x is an odd natural number


where 'x' is a graphical representation of the element


':' stands for 'such that'.


'{}' denotes 'the full set.'


S = x:x is an even prime number and can thus be interpreted as "the set of all x such that x is an even prime number." The roster form for this set S is S = 2. There is only one element in this set. Such sets are referred to as Singleton Sets.

Common Symbols used in Set Theory

A variety of symbols are used to represent common sets. Let's go into detail about each of them.



Corresponding set


It denotes the set of all natural numbers, that is, all positive integers.


Examples: 1, 12, 163, 823 and so on.


It is used to represent the full number set. This symbol is derived from the Greek word 'Zahl,' which means 'number.'


Positive integers are represented by Z+, whereas negative integers are represented by Z-.


Examples: -12, 0, 1237 etc.


It stands for the collection of rational numbers. The word "Quotient" is the basis for the sign. As the quotient of two integers, positive and negative rational numbers are denoted by Q+ and Q-, respectively (with a non-zero denominator).


Examples: 16/17, -6/7 etc.


It is used to represent real numbers as well as any other number that may be stated on a number line.


Positive and negative real numbers are represented by R+ and R-, respectively.


Examples: 2.67, π, 2√3, etc.



It is used to represent a collection of complex numbers.

Examples: 2+ 5i, i, etc.


Other symbols:



Symbol Name







Not a subset

Proper subset

Proper superset


Not superset


Empty set

P (C)

Power set


Equal Set



Element of

Not an element of