Let αbe a recursive function that is injective. We set A=Ran(α) B={x: there exists y>

step 1 There is an injective mapping from A to B. Each element of A has a unique image in B. Okay, sinc

Let αbe a recursive function that is injective. We set ...

Let $\alpha$ be a recursive function that is injective. We set $$ \begin{aligned} &A=\operatorname{Ran}(\alpha) \\ &B=\{x: \text { there exists } y>x \text { such that } \alpha(y)<\alpha(x)\} \end{aligned} $$ (a) Show that $B$ is recursively enumerable and that its complement is infinite. (b) Assume that there exists an infinite recursively enumerable subset $C \subset \mathbb{N}$ that is disjoint from $B$. Show that $A$ is recursive. (c) Show that there exists a recursively enumerable set which has (1) a nonempty intersection with every infinite recursively enumerable set, and (2) an infinite complement.

Let $\alpha$ be a recursive function that is injective. We set
$$
\begin{aligned}
&A=\operatorname{Ran}(\alpha) \\
&B=\{x: \text { there exists } y>x \text { such that } \alpha(y)<\alpha(x)\}
\end{aligned}
$$
(a) Show that $B$ is recursively enumerable and that its complement is infinite.
(b) Assume that there exists an infinite recursively enumerable subset $C \subset \mathbb{N}$ that is disjoint from $B$. Show that $A$ is recursive.
(c) Show that there exists a recursively enumerable set which has (1) a nonempty intersection with every infinite recursively enumerable set, and (2) an infinite complement.

Key Concepts

Recursive (Decidable) Sets

A recursive, or decidable, set is one for which there exists a Turing machine that will correctly decide membership for any given element in a finite amount of time. The distinction between recursively enumerable and recursive sets is key in computability theory, as some sets may be effectively listed without having a total decision procedure.

Infinite Sets and their Complements

The consideration of infinite sets and properties of their complements is a central theme in recursion theory. Showing that the complement of a recursively enumerable set is infinite, or constructing an r.e. set that intersects every infinite r.e. set, involves understanding how enumeration and decision processes interact with notions of infinity and density within the natural numbers.

Recursively Enumerable Sets

Recursively enumerable (r.e.) sets are those for which there exists a Turing machine that will list all their elements, possibly without halting. These sets play a central role in computability theory as they capture the idea of semi-decidability, where membership can be confirmed in finite time even if non-membership might not be decidable.

Recursive Functions

A recursive (or computable) function is one whose value can be computed by a Turing machine for every input in a finite number of steps. In computability theory, these functions form the basis for defining effective procedures and algorithms, and many properties of sets and functions are studied from this perspective.

Injective Functions

An injective function is one that never maps two distinct inputs to the same output. This property is crucial when analyzing the behavior of the function’s range, ensuring that each output uniquely corresponds to one input, which simplifies many arguments in computability regarding the ordering and manipulation of outputs.

Range of a Function

The range of a function is the set of all outputs that the function produces when applied to every element of its domain. In computability, when the function is recursive, its range is a recursively enumerable set, meaning that there is an effective procedure to list its elements, even if the set itself might not be decidable.

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