Set (mathematics)

This article is about sets in mathematics. See also

Set (god) for the figure from ancient Egyptian mythology.
Set game for the card game.

In mathematics, a set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. For a discussion of the properties and axioms concerning the construction of sets, see naive set theory and axiomatic set theory. Here we give only a brief overview of the concept.

Sets are one of the base concepts of mathematics. A set is, more or less, just a collection of objects, called its elements. Standard notation uses braces around the list of elements, as in:

{red, green, blue}

{red, blue, green, red}

{x : x is a primary color}

All three lines above denote the same set. As you see, it is possible to describe one and the same set in different ways: either by listing all its elements (best for small finite sets) or by giving a defining property of all its elements; and it does not matter in what order, or how many times, the elements are listed, if a list is given.

If A and B are two sets and every x in A is also contained in B, then A is said to be a subset of B. Every set has as subsets itself, called the improper subset, and the empty set {}. The union of a collection of sets S = {S₁, S₂, S₃, ...} is the set of all elements contained in at least one of the sets S₁, S₂, S₃, .... The intersection of a collection of sets T = {T₁, T₂, T₃, ...} is the set of all elements contained in all of the sets. The union and intersection of sets, say A₁, A₂, A₃, ... are denoted A₁ ∪ A₂ ∪ A₃ ∪ ... and A₁ ∩ A₂ ∩ A₃ ∩ ... respectively. If you don't mind jumping ahead a bit, the subsets of a given set form a Boolean algebra under these operations. The set of all subsets of X is called its power set and is denoted 2^X or P(X).

Examples of sets of numbers include:

Natural numbers which are used for counting the members of sets.
Integers which appear as solutions to equations like x + a = b.
Rational numbers which appear as solutions to equations like a + bx = c.
Algebraic numbers which can appear as solutions to polynomial equations (with integer coefficients) and may involve radicals.
Real numbers which include transcendental numbers (which can't appear as solutions to polynomial equations with rational coefficents) as well as the algebraic numbers.
Complex numbers which provide solutions to equations such as x² + 1 = 0.

Statistical theory is built on the base of set theory and probability theory.

Care must be taken with verbal descriptions of sets. One can describe in words a set whose existence is paradoxical. If one assumes such a set exists, an apparent paradox or antinomy may occur. Axiomatic set theory was created to avoid these problems.

For example: Suppose we call a set "well-behaved" if it doesn't contain itself as an element. Now consider the set S of all well-behaved sets. Is S well-behaved? There is no consistent answer; this is Russell's paradox. In axiomatic set theory, no set can contain itself as an element, and we do not allow a set of all sets.