Set Theory Quickstart

Set theory is the true foundation of mathematics; they govern basically everything else. A set is, in essence, a collection of objects, and set theory is the study of these collections.

For to be a set, it must be well-defined, i.e., there must be a clear way to determine whether an object is in the set or not.

There are a few ways to define a set:

1. Roster notation: Listing all the elements of the set within curly braces. For example, the set of even numbers less than 10 can be written as .
2. Set-builder notation: Describing the properties that the elements of the set must satisfy. For example, the set of even numbers less than 10 can be written as .

Sometimes we can use a both notations to define a set. For example, the set of all even natural numbers can be written as or .

Recall the sets of numbers that we have learned previously. We will now try to define them more formally using this notation:

Notation and Terminology

Let be the set of even natural numbers. We can write this as or .

Since the number is in the set , we write . If the number is not in the set , we write .

If we have a set and a set , and everything in is also in , we say that is a subset of , and write .

If, additionally, there is at least one element in that is not in , i.e. , we say that is a proper subset of , and write . To prove that , we need to show that and .