# Study About Set Theory Symbols and Types Like Singleton Here

Set Theory is an area of mathematics that studies sets and their properties. A set is defined as a group of items. The elements of a certain set are its objects. Set theory is the study of such sets and the relationships that exist between them. Set theory has proven to be an immensely useful tool for defining some of mathematics’ most complex and crucial structures and therefore is an important part of the numerical ability/quantitative aptitude syllabus in various competitive exams.

On that note, let’s discuss Set Representation, Types 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 Form

**Roster Form**

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

For example, if the set represents all leap years between 1995 and 2015, it would be stated in Roster form as

A ={1996,2000,2004,2008,2012}

**Set Builder Form**

All elements in set builder form share a common property. This property does not apply to objects that do not belong to a set.

For instance, if set S contains all items that are even prime numbers, it is written as

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

where x is an odd natural number

where ‘x’ is a graphical representation used to describe the element

‘:’ denotes ‘such that’.

‘{}’ signifies ‘the entire set.’

As a result, S = { x:x is an even prime number } can be translated as “the set of all x such that x is an even prime number.” S = 2 is the roster form for this set S. This set only has one element. Such sets are known as **Singleton Sets.**

**Types of Sets**

The sets are further classified into many types based on the elements or types of elements. In basic set theory, there are three types of sets:

- Finite set: The total number of elements is limited.
- Infinite set: The number of elements is indefinite.
- Empty set: A set with no elements.
- Singleton set: A set with a single element.
- Equal set: If two sets have the same elements, they are equal.
- Equivalent set: If two sets have the same amount of elements, they are equal.
- Power set: A set of every possible subset.
- Universal set: Any set that contains all of the sets being considered.
- Subset: When all of the elements of set A are elements of set B, A is a subset of B.

**Common Symbols used in Set Theory**

Common sets are represented by a variety of symbols. Let’s discuss them in detail.

Symbol | Corresponding set |

N | It represents the set of all natural numbers, i.e. all positive integers.
Examples: 1, 13, 165, 923 and so on. |

Z | It is used to represent the entire set of numbers. This symbol is taken from the Greek word ‘Zahl,’ which denotes the number.
A set of Positive integers is represented by Z+ and that of negative integers as Z– respectively. Examples: -13, 0, 1025 etc. |

Q | It represents the set of rational numbers. The sign is based on the word ‘Quotient.’ Positive and negative rational numbers are indicated by Q+ and Q– as the quotient of two integers (with a non-zero denominator).
Examples: 17/11, -6/7 etc. |

R | It is used to represent real numbers or any number that can be expressed on a number line.
R+ and R– represent positive and negative real numbers, respectively.
Examples: 2.34, π, 2√7, etc. |

C | It’s used to represent a collection of complex numbers. Examples: 2+ 3i, 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 |