Abstract Set
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An abstract set is an abstract entity that can represent zero or more distinct entities (set members).
- AKA: Unordered Set, Group, Set (Mathematics).
- Context:
- It can range from being an Empty Set to being a Nonempty Set (such as a degenerate set).
- It can range from being a Finite Set (such as a binary set) to being an Infinite Set (such as a formal numerical sequence).
- It can be in a Subset Relation with another Set (e.g. its power set).
- It can be associated with a Set System.
- It can be associated with a Measurable 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 range from being a Labeled Set to being 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.
- a Set of Data Items.
- Counter-Example(s):
- a Singleton, such as a number.
- an Ordered Set, such as a partially ordered set, or a sequence.
- a Multiset (with repeated members).
- a Physical Entity.
- See: Discrete Math, Cluster.
References
2013
- (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Set_(mathematics) Retrieved:2013-12-1.
- In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn Diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The term itself was coined by Bolzano in his work The Paradoxes of the Infinite.
2009
- 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 of several parts; A well-defined collection of mathematical objects and 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 with an infinite number of elements.