An Introduction To Set Theory

Set theory is a branch of logic which studies sets and their properties. A set is a well-defined collection of non-repeating “things” (which could be numbers, people, places etc) which are known as elements of the set.

Set theory is a powerful tool which is able to describe most of mathematics, but is a topic most will only encounter in university. The basics, however, require little mathematical background and can be learned with relative ease.

Terminology

A set is represented as a list of elements between two braces ( $\{\}$ ).
For example, the set containing the whole numbers between 1 and 4 is: $\{1, 2, 3, 4\}$

There are many properties used to describe a set and so before going any further it is important to meet
several of the most common ones.

A finite set is one with a non-infinite number of elements.
An infinite set is one with an infinite number of elements.
The empty set is the set without any elements, commonly represented with the symbol $\emptyset$ .}

A subset of another set is a set whose elements are all also elements of the other set. For example, $\{1,2\}$ is a subset of the set $\{1,2,3,4\}$ but $\{1,2,5\}$ is not as it has an element, $(5)$ , not in $\{1,2,3,4\}$ . A subset is represented using the symbol $\subset$ . Returning to the previous example, we can say: $\{1,2\}\subset\{1,2,3,4\}$ . In general $A\subset B$ says that set A is a subset of set B.

Sets You May Have Encountered

Many are surprised to learn they have already encountered a few sets during their lifetime. These were most likely not referred to as sets but are nonetheless. Below are a few you may have already met:

The integers, sometimes referred to as the “whole” numbers. These are positive and negative numbers without a decimal.
Some examples are -2, -15, 1, 6 etc. These are represented with the symbol $\mathbb{Z}$ .
The natural numbers, sometimes referred to as the “counting numbers”. These are the positive integers represented with the symbol $\mathbb{N}$ . There is some debate as to whether zero should be included in the natural numbers. The natural numbers are a subset of the integers.
There are many more that you may be familar with, such as the rational and irrational numbers, the real numbers (all the previous sets are subsets of the real numbers) and some may also be familiar with the imaginary and complex numbers.

Extra Resources

Here are some extra resources in text and video format if you would like to delve deeper into set theory: