Set Theory is a mathematical branch that deals with sets and their properties. A collection of objects or elements is known as a Set. These objects are frequently referred to as elements or members of a set. A set is, for example, a group of cricket players.

Set theory has numerous real-world applications. From mathematics, topology, and data interpretation to engineering, its use has become very important and dependable. Because set theory is abstract, it is immensely flexible. It aids in the study and analysis of graphs, as well as in understanding the fundamentals of topology. Engineers also utilise it to deal with circuits.

Let’s learn about sets, types of sets, representation, set theory symbols and more with examples.

Table of Contents

## Definition of Sets

A set, as stated in the introduction, is a well-defined collection of objects or people. Many examples of sets in real life include the number of rivers in India, the number of colours in a rainbow, and so on.

## Types of Sets

The elements or types of elements in the sets are used as the basis to classify sets. Various types of sets are

Finite set: A set comprising a limited number of elements.

Infinite set: A set comprising an infinite number of elements.

Empty set: A set with zero elements.

Subset: A is a subset of B when every element of set A is also an element of set B.

Singleton set: A set with only one element.

Equal set: Two sets having the same elements.

Equivalent set: Two sets having an equal number of elements.

Power set: A set of each possible subset.

Universal set: Any set that includes every set being taken into account.

## Representation of Sets

Sets can be represented in two ways:

- Roster Form or Tabular form
- Set Builder Form

### Roster Form

In this method, we list all of the elements of a set, which are enclosed in curly brackets {}. Each element is only written once and is separated by commas.

The set of letters in the word “California,” for example, is expressed as A = {c, a, l, i, f, o, r, n}.

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### Set Builder Form

The set builder form represents all of the elements of a set with a statement or an expression. We will not list the elements in this method; instead, you write the representative element as a variable followed by a vertical line or colon, followed by the general attribute of the same representative element.

For example, in set builder form, the same set (that signifies the set of letters in the word “California”) can be stated as A = {x | x is a letter of the word “California”} or A = x{x : x is a letter of the word “California”}.

## Common Symbols used in Set Theory

A variety of symbols are used to represent sets. Let’s discuss each in depth.

Symbol | Corresponding set |

N | It refers to the set of all natural numbers or positive integers. Examples: 1, 15, 163, 919 and so on. |

Z | It is used to represent the set of integers. A set of positive integers are represented by Z^{+}, whereas a set of negative integers are represented by Z^{–} respectively. Examples: -11, 0, 267 etc. |

Q | It is the set of rational numbers. The word ‘Quotient’ inspired the sign. Q^{+} and Q^{–} represent positive and negative rational numbers as the quotient of two integers (with a non-zero denominator). Examples: 13/17, -2/9 etc. |

R | It is used to represent real numbers as well as any other number on a number line. Positive and negative real numbers are represented by R^{+} and R^{–}, respectively. Examples: 2.45, π, 2√3, etc. |

C | It is used to represent a set of complex numbers. Examples: 4+ 5i, i, etc. |

Other symbols:

Symbols | 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 |

A^{c} | Complement |

∈ | Element of |

∉ | Not an element of |