Set

A Set is an Abstract Entity that can Represent Zero or more Distinct Set Members.

• AKA: Group, Unordered Set, Set (Mathematics).
• Context:
  • It can be:
    • a Empty Set.
    • a Degenerate Set.
    • a Finite Set, such as a Binary Set, Categorical Set, Ordinal Set.
    • an Infinite Set, such as a Formal Numerical Sequence.
  • It can be:
    • an Unordered Set, such as a Categorical Set.
    • an Ordered Set, such as an Ordinal Set.
  • It can be a Member of (in a Subset Relation) a Set (e.g. its Power Set).
  • It can be associated with a Set System.
  • It can be associated with a Measureable Space.
  • It can be in a Set Relation with another Set (e.g. a Distinct Set Relation).
  • It can be the input to a Set Operation, such as a Set Function or Set Relation.
  • It can have a Set Definition, and be:
    • an Explicit Set.
    • an Implicit Set.
  • It can range from being a Small Set to being a Large Set, based on its Set Cardinality.
  • It can be:
    • a Labeled Set.
    • an Unlabeled Set.
• Example(s):
  • The set {3, 4, 1, 2}.
  • The set composed of the first four Whole Numbers.
  • The set {True, False}; a Binary Set known as the Truth Set.
  • The set of all Persons; a Finite Set.
  • The set of all Positive Integers; a Countably Infinite Set).
  • The set of all Sets; an Uncountably Infinite Set.
  • a Concept Class.
  • a Sequence or String.
  • a Tuple or Vector.
  • a Set of Sets.
  • a Set of Multisets.
• Counter-Example(s):
  • a Physical Entity.
  • a Number.
  • a Multiset with repeating Members.
  • a Sequence.
• See: Discrete Math, Cluster, Sequence.

References

• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Set
  • A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. The study of the structure of sets, set theory, is rich and ongoing. ...
  • There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description. See this example:
    • A is the set whose members are the first four positive integers.
  • The second way is by extension, that is, listing each member of the set. An extensional definition is notated by enclosing the list of members in braces:
    • C = {4, 2, 1, 3}
• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Set_(computer_science)
  • In computer science, a set is a collection (container) of certain values, without any particular order, and no repeated values. It corresponds with a finite set in mathematics. Disregarding sequence, and the fact that there are no repeated values, it is the same as a list. ...
• http://en.wiktionary.org/wiki/set
  • A matching collection of similar things; A collection of various objects for a particular purpose; An object made up several parts; A well-defined collection of mathematical objects, its elements, often having a common property; (informal) Set theory; A group of people, usually meeting ...
• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Set_(category_theory)
  • In mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. ...
• http://planetmath.org/encyclopedia/Set.html
  • Sets can be of “real” objects or mathematical objects, but the sets themselves are purely conceptual. This is an important point to note: the set of all cows (for example) does not physically exist, even though the cows do. The set is a “gathering” of the cows into one conceptual unit that is not part of physical reality. This makes it easy to see why we can have sets with an infinite number of elements; even though we may not be able to point out infinitely many objects in the real world, we can construct conceptual sets which an infinite number of elements (see the examples below).