## Definition

A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using set bracket, i.e., { }. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.^{1}

Some example of sets:

- Set of all positive integers, I = {1,2,3,4,5,6,7, …}
- A set of all the planets in the solar system P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
- Set of vowels in English alphabet, A={a,e,i,o,u}
- Set of odd numbers less than 10, B={1,3,5,7,9}

## Who

Have you ever wondered why you learned about sets? They are used everywhere and understanding them is essential to many fields, e.g., mathematics and database programming.

## What

German mathematician G. Cantor introduced the concept of sets. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.^{1}

## Why

Set theory is used throughout mathematics. It is used as a foundation for many subfields of mathematics. In the areas pertaining to statistics, it is particularly used in probability. Much of the concepts in probability are derived from the consequences of set theory. Indeed, one way to state the axioms of probability involves set theory.^{2}

Database programming. In fact, as a programmer, I use mathematical containers a lot. Bags, Sets, Mappings, Queues, Stacks, etc.

Sets are mathematically the smallest (simplest) container-like structure I know that can be used to describe everything I do for proof purposes (as most mathematicians will find). They are not an end-all-be-all (I find Types to be closer, but still lacking); however, sets are useful whenever I need to work with a situation that asks a question: “Is this included or not?”.

Furthermore, sets and set theory are really a starting point to proof-based math, and more specifically, understanding set theory is a foundation to combinatorics, which is a foundation to probability theory, which is a foundation to statistics.

Therefore, for most people who use statistics, the real life application of set theory is to prepare them for statistical analysis. Of course statistics works on huge collections of data, and you cannot understand the data without understanding the collections.

In general though, most mathematical fields have been built up from set theory first. That includes building up proofs as to why addition of numbers is commutative and associative, etc.

Nicholas Cooper^{3}

Set Theory is the foundation of many aspects of Computer Systems Engineering and data management.

- Binary Logic gates used to build microchips in ever digital device.
- ‘Selection’ one of the three fundamental constructs of Programming (the other two being sequence & iteration).
- Sets form the basis of many data structure used in programming, e.g. Set (Java Platform SE 7).
- Databases, Set Theory determines which data will be included and exclude in searches and selections.
- Security & Access control lists for users and systems.
- Distribution lists for messages such as email.
- Cryptography for secure communications.
- Bayesian filtering of junk email.
- Artificial Intelligence decision making.
- Image Recognition.
- Linguistics, Translation and proto-language study.

Martin Spamer, BSc Computer Science & Software Engineering, Staffordshire University^{3}

See Theoretical Knowledge Vs Practical Application.

## How

I don’t show you the basics of sets. Many of the **References** and **Additional Reading** websites and **Videos** will assist you with that. As some professors say: “It is intuitively obvious to even the most casual observer.”

## References

^{1} “Discrete Mathematics – Sets”. 2021. *tutorialspoint.com*. https://www.tutorialspoint.com/discrete_mathematics/discrete_mathematics_sets.htm.

^{2} “The Fundamental Cioncept Of Set Theory”. 2021. *thoughtco*. https://www.thoughtco.com/what-is-set-theory-3126577.

^{3} “What Are Some Real-Life Applications Of Set Theory?”. 2021. *Quora*. https://www.quora.com/What-are-some-real-life-applications-of-set-theory.

## Additional Reading

Falcão, João Renato. “Set Theory — From Pure Math To SQL”. 2022. *Medium*. https://falcaojoaorenato.medium.com/math-concepts-for-sql-programming-232cb1da0d16.

This article intends to explain the fundamental theory applied in SQL, probability and statistics.

“FREE Step-By-Step Types Of Sets Lesson With Interactive Exercises | Math Goodies”. 2021. *mathgoodies.com*. https://www.mathgoodies.com/lessons/sets/types-of-sets.

“Introduction To Sets”. 2021. *mathsisfun.com*. https://www.mathsisfun.com/sets/sets-introduction.html.

“Math: Sets & Set Theory (Video Lessons, Examples And Solutions)”. 2021. *onlinemathlearning.com*. https://www.onlinemathlearning.com/math-sets.html.

“Mathematics | Introduction Of Set Theory – GeeksForGeeks”. 2015. *geeksforgeeks*. https://www.geeksforgeeks.org/set-theory/?ref=lbp.

“Probability | Statistics And Probability | Math | Khan Academy”. 2021. *Khan Academy*. https://www.khanacademy.org/math/statistics-probability/probability-library#basic-set-ops.

“Set (Mathematics) – Wikipedia”. 2021. *en.wikipedia.org*. https://en.wikipedia.org/wiki/Set_(mathematics).

“Set — From Wolfram Mathworld”. 2021. *mathworld.wolfram.com*. https://mathworld.wolfram.com/Set.html.

“Set Symbols”. 2021. *mathsisfun.com*. https://www.mathsisfun.com/sets/symbols.html.

“Set Theory | Basic Concepts Of Set Theory – hitbullseye”. 2023. *hitbullseye.com*. https://www.hitbullseye.com/Quant/Set-Theory.php.

“Set Theory | Introduction To College Mathematics”. 2021. *courses.lumenlearning.com*. https://courses.lumenlearning.com/atd-hostos-introcollegemath/chapter/set-theory/.

“Sets & Set Theory”. 2021. *storyofmathematics.com*. https://www.storyofmathematics.com/sets-set-theory.

“Types Of A Set”. 2021. *tutorialspoint.com*. https://www.tutorialspoint.com/types-of-a-set.

## Videos

This video explains “Introduction to Set Theory and Representation of a Set”. Watch the full video for complete explanation.

An explanation of the branch of logic known as set theory which deals with groups of objects and serves as the foundations of mathematics. This series covers the basics of set theory and higher order logic. In this month we are looking at the properties of sets and classes, including transitive sets, swelled sets, supercomplete sets, ordinary sets, proper subsets, null sets, empty sets, universal sets, and void sets. We are also looking at the first four axioms of a basic universe, following Neumann Berneays Gödel (NBG) set theory.