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I am studying for finals and I came across this question:

If $A$ and $B$ are denumerable, then $A-B$ is denumerable.

I saw this had been asked in the past but there was not a concrete answer as to whether this can be proved or not.

I would think $A-B$ would still be denumerable but not sure since denumerable means the set is equivalent to the natural numbers.

Pedro
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Sam
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    What happens when $A=B$? – Asinomás Dec 13 '16 at 21:25
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    "since denumerable means the set is equivalent to the natural numbers" is incorrect. A set is said to be denumerable (also called countable) when it is in bijection with the natural numbers or some subset of the natural numbers. Finite sets are denumerable as well. Notice that $A-B$ is a subset of $A$. – JMoravitz Dec 13 '16 at 21:25
  • @JMoravitz yes well for me equivalence means bijection exists – Sam Dec 13 '16 at 21:26
  • As $A$ is denumerable, you may as well assume that it is a subset of $\Bbb N$. But then so is $A-B$. – Hagen von Eitzen Dec 13 '16 at 21:26
  • No bijection exists between ${1}$, ${1,2,3}$ and $\Bbb N$ yet they are all denumerable – JMoravitz Dec 13 '16 at 21:26
  • Are finite sets denumerable then? – Sam Dec 13 '16 at 21:27
  • @Sam, that's what JMoravitz wrote in their first comment above. – avs Dec 13 '16 at 21:27
  • @avs oh oops, couldn't see it well since I am on a tablet – Sam Dec 13 '16 at 21:28
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    @JMoravitz ""since denumerable means the set is equivalent to the natural numbers" is incorrect " is incorrect. Sometimes "denumerable" includes only infinite sets (see this and this). The meaning (and thus the answer of the question) depends on the chosen definition. – Pedro Dec 13 '16 at 21:47
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    @JMoravitz I have never seen "denumerable" used to mean "integers or smaller"! I always believed that it was introduced specifically to create an unambiguous term to deprecate the use of "countable" for "countably infinite". I imagine that many learners find it odd to say that ${1,2,3}$ is "not countable", given how easy it is to literally count that set, and hence it is often preferable to use "countable" to mean "integers or smaller", making it the exact complement of "uncountable". – Erick Wong Dec 13 '16 at 21:53
  • The proof of the Lemma is here https://math.stackexchange.com/q/107617/837396 –  Dec 06 '20 at 14:04

1 Answers1

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Lemma: A subset of a countable (denumerable) set is countable.

Proof of lemma: Let $A$ be a countable set and $B\subseteq A$. Since A is countable there is an injection $f:A\to \mathbb{Z^+}$. Now the restriction of $f$ to $B$ is an injection from $B$ to $\mathbb{Z^+}$. Hence $B$ is countable. $\Box$

Now it is required to prove that $A-B$ is countable (denumerable). Obviously $A-B\subseteq A$. Since $A$ is countable, by lemma, it follows that $A-B$ is countable. $\Box$

Janitha357
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  • I would think you’d have to exhibit a bijection h:A$\mapsto A\setminus B$ –  Dec 06 '20 at 14:07