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[This is a duplicate of my question from Mathematics Stack Exchange:

https://math.stackexchange.com/questions/4792354/many-one-equivalence-of-sets-that-differ-finitely

I am posting it here since it didn't get an answer there.]

Consider two sets of natural numbers, $A$ and $B$. Suppose that they differ finitely, i.e., their symmetric difference $(A-B)\cup (B-A)$ is a finite set. Is it true that $A$ and $B$ are many-one equivalent?

According to Odifreddi (Exercise III.2.2. f) in his "Classical recursion theory Vol. 1") and Rogers (Exercise 7-4 in his "Theory of recursive functions and effective computability", under additional assumption that $A$ and $B$ are infinite) the sets $A$ and $B$ are many-one equivalent.

However, taking $A=\mathbb{N}-\{0\}$ and $B=\mathbb{N}$, $A$ and $B$ clearly differ finitely, but are not many-one equivalent.

Am I not seeing something important, or both Odifreddi and Rogers gave false statements as exercises?

(It seems that if $A$ and $B$ are nonempty and not equal to the whole set of natural numbers then indeed they are many-one equivalent if they differ finitely. However, the question still stands since I don't see why both Odifreddi and Rogers would give false statements, so I presume that I am missing something here.)

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    $\begingroup$ You are right. $ $ $\endgroup$ Nov 22, 2023 at 8:24
  • $\begingroup$ What is the definition of many-one equivalent? $\endgroup$
    – domotorp
    Nov 23, 2023 at 21:05
  • $\begingroup$ @domotorp I think it's: $A$ many-to-one reducible to $B$ iff there is a many-to-one reduction from $A$ to $B$, i.e., a computable function $f$ such that $∀ x, x ∈ A ⟺ f(x) ∈ B$; and $A$ many-to-one equivalent to $B$ iff $A$ many-to-one reducible to $B$ and $B$ many-to-one reducible to $A$. $\endgroup$ Nov 24, 2023 at 9:33
  • $\begingroup$ And where is the problem? That $f$ needs to be $\mathbb N\to \mathbb N$? $\endgroup$
    – domotorp
    Nov 24, 2023 at 15:19

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