Properties
If A is a recursive set then the complement of A is a recursive set. If A and B are recursive sets then A ∩ B, A ∪ B and the image of A × B under the Cantor pairing function are recursive sets.
A set A is a recursive set if and only if A and the complement of A are both recursively enumerable sets. The preimage of a recursive set under a total computable function is a recursive set. The image of a computable set under a total computable bijection is computable.
A set is recursive if and only if it is at level Δ0
1 of the arithmetical hierarchy.
A set is recursive if and only if it is either the range of a nondecreasing total computable function or the empty set. The image of a computable set under a nondecreasing total computable function is computable.
Read more about this topic: Recursive 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)