Recursive Set - Properties

Properties

If A is a recursive set then the complement of A is a recursive set. If A and B are recursive sets then AB, AB 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.

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