Let f ∈ℱ1 . Show that f is recursive if and only if its graph G={(x, y) ∈ℕ^2: y=f(x)} is recursi

step 1 First part of this, we have to show that this function is even. So I'm going to plug in half of

Let f ∈ℱ1 . Show that f is recursive if and only if its...

Key Concepts

Recursive Function

A recursive function is a function for which there exists an algorithm that, given any input from its domain, produces the correct output in a finite number of steps. This is a central concept in computability theory, as it formalizes the notion of effective computability and defines the class of functions that can be computed by mechanical procedures.

Recursive Set (Decidable Set)

A recursive set, or decidable set, is a set for which there is an algorithm that can decide in finite time whether any given element belongs to it. This concept is fundamental in computability and logic; it addresses the idea of algorithmic decidability and is essential when discussing the computability of various mathematical objects.

Graph of a Function

The graph of a function is the set of all ordered pairs (x, y) such that y equals the value of the function at x. In the context of computability theory, examining the graph of a function as a set helps in analyzing whether the function itself is computable by determining if there exists an algorithm that can, for any given pair (x, y), decide whether y is indeed the output of the function for x.

Equivalence of Function Recursiveness and Graph Recursiveness

This concept captures the relationship that a function is computable (or recursive) if and only if its graph is a decidable (or recursive) set. This equivalence is a key insight in computability theory, linking the process of function computation with the decidability of the set of input-output pairs, thus providing a powerful method to study and characterize recursive functions.

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