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 Basic Set Theory and Set theory. Here we give only a brief overview of the concept.

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.

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} {x:xis a primary color}

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.

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* = {*S*_{1}, *S*_{2}, *S*_{3}, ...} is the set of all elements contained in at least one of the sets *S*_{1}, *S*_{2}, *S*_{3}, ...}
The intersection of a collection of sets *T* = {*T*_{1}, *T*_{2}, *T*_{3}, ...} is the set of all elements contained in all of the sets. The union and intersection of sets, say
*A*_{1}, *A*_{2}, *A*_{3}, ... are denoted
*A*_{1} u *A*_{2} u *A*_{3} u ...
and
*A*_{1} n *A*_{2} n *A*_{3} n ...
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 + b
*x*= 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
*x*^{2}+ 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.