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.

 

Symbol	Corresponding set
N	It denotes the set of all natural numbers, that is, all positive integers.   Examples: 1, 12, 163, 823 and so on.
Z	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.
Q	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.
R	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.
C	It is used to represent a collection of complex numbers. Examples: 2+ 5i, i, etc.

 

Other symbols:

 

Symbols	Symbol Name
{}	Set
U	Union
∩	Intersection
⊆	Subset
⊄	Not a subset
⊂	Proper subset
⊃	Proper superset
⊇	Superset
⊅	Not superset
Ø	Empty set
P (C)	Power set
=	Equal Set
Ac	Complement
∈	Element of
∉	Not an element of