Set Theory: Formulas & Examples with Basics

Georg Cantor - the founder of set theory
Georg Cantor – the founder of set theory

“Set Theory” is a branch of mathematics that studies sets, which are made up of the collection of distinct and well-defined objects called elements or members of that particular set.

George Cantor created this theory was born in Russia on March 3, 1845. In 1873, he published an article that marks the birth of a set theory. George Cantor died in Germany on January 6, 1918.

the terminology “set theory” is often used in mathematical-related objects only.

before we go into the details of set theory, let us take a brief introduction, purpose, and importance along with some examples of sets in our surroundings.

a typical set theoryVenn Diagrams to Express a set
a typical set theory Venn Diagram to Express a set.  In our daily life, we use the word set only for some particular collections such as a water set, tea set, dinner set, sofa set, set of books, a set of colors, and so on.


in mathematics, the word set has broader meanings than those in our daily life because it provides us with a way to integrate the different branches of mathematics. It also helps to solve many mathematical problems of both simple and complex nature.

In short,

it plays a pivotal role in the advanced study of mathematics in the modern age, which shows the importance of set theory. Look at the following examples of a set.

  1. A= The set of counting numbers.
  2. B= The set of American States.
  3. C= The set of Geometrical Instruments.

A set cannot consist of elements like moral values, concepts, evils or virtues, etc.
“A set is a collection of well-defined and distinct objects/numbers. The objects/numbers in any set are called its members or elements”.

Expressing a Set.

There are three ways to express a set. this means there are three types of a set which are as follows:

1. Descriptive Form        2. Tabular Form       3. Set Builder Form

Descriptive Form

If a set is described with the help of a statement, it is called descriptive form. For example:

  1. N = set of Natural Numbers
  2. Z = set of Integers
  3. P = set of Prime Numbers
  4. W = set of Whole Numbers
  5. E = Set of Even Numbers
  6. O = Set of Odd Numbers
  7. S = set of solar months start with the letter “J”

Do You Know?

the sets of natural numbers , whole numbers, integers, even numbers and odd numbers are denoted by  the English letters N,W,Z,E and O respectively.

Tabular Form

If we list all the elements of a set within the braces { } and separate each element by using a comma”, ”  it is called the tabular or roster form.

for example:

  1. A= { a,e, i,o,u}
  2. M={football, hockey, cricket, squash, soccer}
  3. W={0,1,2,3……..}
  4. C={3,6,9,………99}
  5. N={0,1,2,3,4,…}
  6. X={a,b,c,…..z}

Set Builder Form

if we describe a set by using a common property of all its elements, it is known as set builder form. A set can also be expressed in set builder form. for example, ” E is a set of even numbers” is the descriptive form, whereas, E= {2,4,6,…} is the tabular form of the same set.

This set in set-builder form can be written as;

E = { x | x is an even number }

and we can read it as E is a set of elements x, such that x is an even number.

  1. A = { x  | x is a solar month of  a year}
  2. B = { x  | x ∈ N  ∧ 1 < x <5}
  3. C = { x  | x ∈ W ∧ x ≤ 4}

Here are Some Important Symbols of Set Theory:

  1. “| ” stand for “such that”
  2. “∧” symbolizes “and ”
  3. “≤” exemplify “less than or equal to”
  4. “≥” implies” greater than or equal to”
  5. “∈” represents ” belongs to”
  6. “∨” appear for  “or “

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