Set

This article is about sets in mathematics. See also

• Set (god) for the figure from ancient Egyptian mythology.
• Set game for the card game.
• Set (computer science) as datatype that is a collection of values.
• set (drama) discusses dramatic sets

In mathematics, a set is a collection of objects such that two sets are equal if, and only if, they contain the same objects. A finite set is a collection of a finite number of objects; the alternative is an infinite set. 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 basic 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, red, blue, red, green, red, red, green, red, red, blue}

{x : x is an additive 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.

Set Terminology

If <math>A</math> and <math>B</math> are sets and every <math>x</math> in <math>A</math> is also contained in <math>B</math>, then <math>A</math> is said to be a subset of <math>B</math>, denoted <math>A \subseteq B</math>. If atleast one element in <math>B</math> is not also in <math>A</math>, <math>A</math> is called a proper subset of <math>B</math>, denoted <math>A \subset B</math>. Every set has as subsets itself, called the improper subset, and the empty set {} or <math>\emptyset</math>. The fact that an element <math>x</math> belongs to the set <math>A</math> is denoted <math>x \in A</math>.

The union of a collection of sets <math>S = {S_1, S_2, S_3, \cdots}</math> is the set of all elements contained in at least one of the sets <math>S_1, S_2, S_3, \cdots</math>

The intersection of a collection of sets <math>T = {T_1, T_2, T_3, \cdots}</math> is the set of all elements contained in all of the sets.

These unions and intersections are denoted

<math>S_1 \cup S_2 \cup S_3 \cup \cdots</math>

and

<math>T_1 \cap T_2 \cap T_3 \cap \cdots</math>

respectively.

The "number of elements" in a certain set is called the cardinal number of the set and denoted <math>|A|</math> for a set <math>A</math> (for a finite set this is an ordinary number, for an infinite set it differentiates between different "degrees of infiniteness", named <math>\aleph_0</math> (aleph zero), <math>\aleph_1, \aleph_2 ...</math>).

The set of all subsets of <math>X</math> is called its power set and is denoted <math>2^X</math> or <math>P(X)</math>. This power set is a Boolean algebra under the operations of union and intersection.

The set of functions from a set A to a set B is sometimes denoted by B^A. It is a generalisation of the power set in which 2 could be regarded as the set {0,1} (see natural number).

The cartesian product of two sets A and B is the set

A×B={(a,b) : a ∈ A and b ∈ B}.

The sum of two sets A and B is the set

A+B = A×{0} ∪ B×{1}.

Examples of Sets of Numbers

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

Special Remarks About Terminology

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 itself well-behaved? There is no consistent answer; this is Russell's paradox. In axiomatic set theory, the set S is not allowed, and we have no paradox.