Simple set
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In recursion theory a simple set is an example of a set which is recursively enumerable but not recursive.
[edit] Definition
A subset S of the natural numbers N is called simple if it satisfies the following properties
- N\S is infinite and contains no infinite recursively enumerable set
- S is recursively enumerable.
An equivalent condition to 1 above is that S ∩ X ≠ ø for any infinite recursively enumerable set X. The complement of a simple set is known as an immune set. There are immune sets whose complements are also immune; these sets are called bi-immune.
[edit] Properties
- The set of simple sets and the set of creative sets are disjoint. A simple set is never creative and a creative set is never simple.
- The collection of sets that are simple or cofinite forms a filter in the lattice of recursively enumerable sets.
[edit] References
- Robert I. Soare, Recursively Enumerable Sets and Degrees, Springer-Verlag, 1987. ISBN 0-387-15299-7
