Set
Elementary
A set is a collection of different objects, called its elements. Order and repetition do not matter: $\setdef{1, 2}$, $\setdef{2, 1}$, and $\setdef{1, 1, 2}$ are the same set.
Graduate (axiomatic set theory)
A set is an object of a set theory such as ZFC. By the axiom of extensionality, a set is determined entirely by its elements.
Examples
Examples
- $\setdef{1, 2, 2}$ Elementary
Non-examples
Things that may be mistaken for examples.
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The roots of $(x - 1)^2$ counted with multiplicity
High school
The roots of $(x - 1)^2$ counted with multiplicity does not form a set, which cannot count an element twice.
- $(1, 2, 2)$ High school
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The class of all sets
Graduate
The class of all sets is not a set, because treating it as one leads to Russell’s paradox.
Properties
- Not every collection is a set: the collection of all sets is not a set (Russell’s paradox).
Distinctions involving this term
- Types of collections: Mathematics has several kinds of “a bunch of things,” which differ in whether order and repetition matter. A set ignores order and repetition. A multiset keeps repetition. A sequence keeps order and repetition. “Collection” is informal, and “group” is a technical term for a set with an operation.