Set (mathematics)

Sets are formally defined in a branch of mathematics known as set theory. Informally, a set is thought of as any collection of distinct elements.

Introduction

Certain restrictions are usually imposed on what may be called a set, though. Commonly one requires that no set may be an element of itself.

A set is not required to have any structure, in particular there's no requirement that the elements have any natural ordering or any properties of any kind at all, except the property of being a member of the set.

Despite the intuitive definition, a set is usually not defined formally in terms of other mathematical objects; rather it is accepted that a set is an "undefined" entity. Because of this property, sets are fundamental structures in mathematics and logic. Mathematicians have found ways to define many mathematical objects, such as the real numbers, in terms of sets.

The number of elements that a set contains does not have to be finite. Sets that contain a finite number of elements are called finite sets. Sets that contain an infinite number of elements are called infinite sets. The number of elements that a finite set contains is called that set's cardinality. The concept of cardinality can also be applied to infinite sets, though the concept is less intuitive, and relies upon bijections between sets.

Notation

Some sets can be denoted by a list of objects separated with commas, enclosed with curly brackets. For example, {1, 2, 3} is the set of the numbers 1, 2, and 3. We say that 1, 2, and 3 are its members.

There are many other ways to write out sets. For example,

A = {x | 1 < x < 10, x is a natural number}

can be read as follows: A is the set of all x, where x is between 1 and 10, and x is a natural number. A could also be written as:

A = {2, 3, 4, 5, 6, 7, 8, 9}

Membership in a set is expressed with the ∈ symbol. To say that the set A contains the 2 as an element (or that 2 is an element of A), we write

2 ∈ A

The cardinality of a set is expressed by placing bars around the name of the set. For example, one would express the cardinality of the above set as such:

|A| = 8

Some special sets

Some sets that are ubiquitous in the mathematical literature have special symbols:

• $\emptyset$, the empty set, sometimes written {}.
• $\mathbb {N}$, the set of natural numbers
• $\mathbb {Z}$, the set of integers
• $\mathbb {Q}$, the set of rational numbers
• $\mathbb {R}$, the set of real numbers

Among other such well known sets are the complex numbers, quaternions, octonions and the hamiltonian integers.

Some examples of sets

• The set consisting of all tuples (a,b), where a is any real number and ditto for b. This set is known as $\mathbb {R}$x$\mathbb {R}$ or $\mathbb {R}$².
• The three element set {Red, Yellow, Green}.
• The set consisting of the two elements Brake, Accelerate.
• The set consisting of all tuples (a,b) where a is any element in the set {Red, Yellow, Green} and b is any element in the set {Brake, Accelerate}.
• The set of all functions from the set {Red, Yellow, Green} to the set {Brake, Accelerate}.

See also

• Set theory
• Naive set theory
• Zermelo-Fraenkel axioms
• Peano axioms
• Mathematics

Related topics

• Cardinal number
• Transfinite algebra
• Aleph-0
• Continuum hypothesis
• Ernst Zermelo
• Thoralf Skolem
• Georg Cantor

External links

• mathworld