In mathematics, a set can be thought of as any collection of distinct things considered as a whole. Though a simple idea, it is nevertheless one of the most important and fundamental concepts in modern mathematics, and the study of the structure of possible sets, set theory, is quite rich.

Set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as primary school. It is the language in which modern mathematics is described. Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived.

This article gives a brief and basic introduction to what mathematicians call "intuitive" or "naive" set theory; for a more detailed account see naive set theory. For a rigorous modern axiomatic treatment of sets see axiomatic set theory.

Definition

---

A set is a collection of objects considered as a whole. The objects of a set are called elements or members. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters, A, B, C, etc. Two sets A and B are said to be equal, written A = B, if they have the same members.

As opposed to a multiset and a real-life collection, a set cannot contain multiple copies of an element.

Describing sets

---

Not all sets have precise descriptions of any sort; they may simply be arbitrary collections, with no expressible "rule" saying what elements are in or out.

Some sets may be described in words, for example:

A is the set whose members are the first four positive whole numbers.

B is the set whose members are the colors of the French flag.

By convention, a set can also be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces), for example:

C = {4, 2, 1, 3}

D = {red, white, blue}

Two different descriptions may define the same set. For example, for the sets defined above, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality. Similarly, for the sets defined above, B = D.

Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example, {6, 11} = {11, 6} = {11, 11, 6, 11}.

For sets with many elements, an abbreviated list is sometimes used. For example, the first one thousand positive whole numbers can be described using the symbolic shorthand:

{1, 2, 3, ..., 1000},

where the ellipsis (...) indicates that the list continues in the obvious way.

Similarly the set of even numbers can be described by the notation:

{2, 4, 6, 8, ... }.

More complicated sets are sometimes described by a different notation. For example the set F, whose members are the first twenty numbers which are four less than a square integer, can be described using the following:

F = {$n^2$ – 4 : n is an integer; and 0 ≤ n ≤ 19}

In this description, the colon (:) means "such that", and the mathematician interprets this description as F is the set of numbers of the form $n^2$ – 4, such that n is a whole number in the range from 0 to 19 inclusive. (Sometimes the pipe notation | is used instead of the colon.)

For more information on describing sets see Set-builder notation.

Set membership

---

If something is or is not an element of a particular set then this is symbolised by $\backslash in$ and $\backslash notin$ respectively. So, for example, with respect to the sets defined above:

• $4\; \backslash in\; A$ and $285\; \backslash in\; F$ (since 285 = 17² − 4); but

• $9\; \backslash notin\; F$ and $\backslash mathrm\{green\}\backslash \; \backslash notin\; B$.

For more information on set membership see Element (mathematics).

Cardinality of a set

---

Each of the sets described above has a definite number of members; for example, the set A has four members, while the set B has three members.

A set can also have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. For example, the set A of all three-sided squares has zero members, and thus A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics.

For more information on the empty set see Empty set.

A set can also have an infinite number of members; for example, the set of natural numbers is infinite.

For more information on infinity and the size of sets, see cardinality and cardinal number.

For more information on finite sets and counting them, see combinatorics and permutations and combinations.

Subsets

---

If every member of the set A is also a member of the set B, then A is said to be a subset of B, written $A\; \backslash subseteq\; B$, also pronounced A is contained in B. Equivalently, we can write $B\; \backslash supseteq\; A$, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by $\backslash subseteq$ is called inclusion or containment.

If A is a subset of but not equal to B, then A is called a proper subset of B, written $A\; \backslash subset\; B$ (A is a proper subset of B) or $B\; \backslash supset\; A$ (B is proper superset of A). However, in some literature these symbols are read the same as $\backslash subseteq$ and $\backslash supseteq$, so it's often preferred to use the more explicit symbols $\backslash subsetneq$ and $\backslash supsetneq$ for proper subsets and supersets.

A is a subset of B