Properties
If A and B are recursively enumerable sets then A ∩ B, A ∪ B and A × B (with the ordered pair of natural numbers mapped to a single natural number with the Cantor pairing function) are recursively enumerable sets. The preimage of a recursively enumerable set under a partial recursive function is a recursively enumerable set.
A set is recursively enumerable if and only if it is at level of the arithmetical hierarchy.
A set is called co-recursively enumerable or co-r.e. if its complement is recursively enumerable. Equivalently, a set is co-r.e. if and only if it is at level of the arithmetical hierarchy.
A set A is recursive (synonym: computable) if and only if both A and the complement of A are recursively enumerable. A set is recursive if and only if it is either the range of an increasing total recursive function or finite.
Some pairs of recursively enumerable sets are effectively separable and some are not.
Read more about this topic: Recursively Enumerable Set
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)