Set Theory Basics

Core Idea

A set is a collection of distinct elements. Sets can be finite or infinite, described by membership rules. Set theory provides foundational language for mathematics using notation for membership (∈), equality, and containment (⊆).

Explainer

A set is one of the most primitive ideas in mathematics: it is simply a collection of distinct objects, called elements, treated as a single thing. The elements can be anything — numbers, letters, people, other sets. What matters is only whether something is in the collection or not. We write this with curly braces: {2, 4, 6} is the set containing 2, 4, and 6. The symbol ∈ means "is an element of," so 4 ∈ {2, 4, 6} is true, while 5 ∈ {2, 4, 6} is false. Its negation, ∉, means "is not an element of."

Two features of sets are worth internalizing early. First, sets are unordered: {1, 2, 3} and {3, 1, 2} are the same set, because they contain exactly the same elements. Second, sets have no duplicates: if you write {1, 2, 2, 3}, the extra 2 is ignored — the set is just {1, 2, 3}. Both properties follow from the fact that membership is a yes-or-no question: either an element belongs to the set or it does not, and listing it twice does not change that answer.

Sets can be described in two ways. Roster notation lists the elements explicitly: {2, 4, 6, 8}. Set-builder notation describes them by a rule: {x | x is an even positive integer less than 10}, read "the set of all x such that x is an even positive integer less than 10." Set-builder notation is essential for infinite sets, which cannot be listed in full.

Containment between sets is captured by the subset relation. We say A ⊆ B (A is a subset of B) if every element of A also belongs to B. The empty set ∅ — the set with no elements — is a subset of every set, because there are no elements in it that could fail to be in B. A proper subset, written A ⊂ B, requires additionally that A ≠ B: B has at least one element that A lacks. These relations are the building blocks for comparing and relating sets, and they will appear throughout mathematics wherever structure is described in terms of membership.