A set is an unordered collection of distinct objects. The objects in a set are typically referred to as its elements. A set is usually denoted by listing all of its elements between braces. For example, \(\{1, 3, 2\}\) denotes a set of three numbers, and because sets are unordered, \(\{3, 2, 1\}\) denotes the same set.\(\{1, 2, 2, 3, 3, 3\}\) does not denote a set, because the elements of a set must be distinct.

Another way to denote sets is the so-called abstraction method, in which the members of a set are given an explicit description, leaving no need for listing them. For example, the set from the example above \(\{1, 3, 2\}\) can be described as the set of all natural numbers \(x\) which are less than \(4\). Formally that is denoted as \(\{x \mid (x < 4) \text{ and } (x \text{ is a natural number})\}\).

I am new to using Latex for editing, so I am not sure if this is the best way to display what’s above. Also, “x is a natural number” is the same as x∈N but we did not get to the membership operator yet.

Using the abstraction method allows for denoting sets with infinitely many elements, which would be impossible by listing them all. For example, the set \(\{x \mid x = 2n \text{ for some natural } n \}\) is the set of all even numbers.

A set doesn’t need to contain things of the same type, nor does it need to contain things that can all be brought to the same place: We could define a set \(S\) that contains the apple nearest you, the left shoe which you wore last, the number 17, and London (though it’s not clear why you’d want to). Rather, a set is an arbitrary boundary that we draw around some collection of objects, for our own purposes.

The use of sets is that we can manipulate representations of sets and study relationships between sets without concern for the actual objects in the sets. For example, we can say “there are 35 ways to choose three objects from a set of seven objects” regardless of whether the objects in the set are apples, people, or abstract concepts. noteThis is an example of abstracting over the objects.

It’s also worth noting that a set of elements is itself a single distinct object, different from the things it contains, and in fact, one set can contain other sets among its elements. For example, \(\{1,2,\{1,2\}\}\) is a set containing three elements: \(1\), \(2\) and \(\{1,2\}\).


  • \(\{1,5,8,73\}\) is a set, containing numbers \(1\), \(5\), \(8\) and \(73\);

  • \(\{\{0,-3,8\}\}\) is a set, containing one element — the set \(\{0,-3,8\}\);

  • \(\{\text{Mercury}, \text{Venus}, \text{Earth}, \text{Mars} \}\) is a set of four planets;

  • \(\{x \mid x \text{ is a human, born on 01.01.2000} \}\) is a set all people whose age is the same as the current year number;

  • \(\{\text{author's favorite mug}, \text{Arbital's main page}, 73, \text{the tallest man born in London}\}\) is a set of four seemingly random objects.

Set membership

Main page: Set membership

Set membership can be stated using the symbols \(∈\) and \(∉\). They describe the contents of a set.\(∈\) indicates what is in a set. \(∉\) indicates what is not in a set. For example, “$x ∈ A$” translated into English is “$x$ is a member of the set \(A\).” and “$x ∉ A$” translates to “$x$ is not in the set \(A\).”

Set cardinality

Main page: Cardinality

The size of a set is called its cardinality. If \(A\) is a finite set then the cardinality of \(A\), denoted \(|A|\), is the number of elements \(A\) contains. When \(|A| = n\), we say that \(A\) is a set of cardinality \(n\). There exists a bijection from any finite set of cardinality \(n\) to the set \(\{0, ..., (n-1)\}\) containing the first \(n\) natural numbers. We can generalize this idea to infinite sets: we say that two infinite sets have the same cardinality if there exists a bijection between them. Any set in bijective correspondence with \(\mathbb N\) is called countably infinite, while any infinite set that is not in bijective correspondence with \(\mathbb N\) is call uncountably infinite. All countably infinite sets have the same cardinality, whereas there are multiple distinct uncountably infinite cardinalities.

See also


  • Set builder notation
  • Cardinality

    The “size” of a set, or the “number of elements” that it has.

  • Convex set

    A set that contains all line segments between points in the set

  • Operations in Set theory
  • Cantor-Schröder-Bernstein theorem

    This theorem tells us that comparing sizes of sets makes sense: if one set is smaller than another, and the other is smaller than the one, then they are the same size.

  • Disjoint union of sets

    One of the most basic ways we have of joining two sets together.

  • Empty set

    The empty set does what it says on the tin: it is the set which is empty.

  • Set product

    A fundamental way of combining sets is to take their product, making a set that contains all tuples of elements from the originals.

  • Finite set

    A finite set is one which is not infinite. Some of these are the least complicated sets.

  • Extensionality Axiom

    If two sets have exactly the same members, then they are equal


  • Mathematics

    Mathematics is the study of numbers and other ideal objects that can be described by axioms.