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.
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}
See also
Related topics
- Cardinal number
- Transfinite algebra
- Aleph-0
- Continuum hypothesis
- Ernst Zermelo
- Thoralf Skolem
- Georg Cantor