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I'm having trouble in trying to understand about the real meaning of countable sets. Can an empty set be considered as a countable set,since there is no element how can we count? But there is a book I am using, it just declare that an empty set is countable without verification. Is there anyone who can help me with this one?

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    $\begingroup$ Do you know the definition of countable set? $\endgroup$
    – Git Gud
    Mar 12, 2015 at 14:14
  • $\begingroup$ Yes I know the one which says a countable set is countable if there exist a 1-1 corresponding between an elements of a set with any subset of natural numbers $\endgroup$
    – Uncool
    Mar 12, 2015 at 14:24
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    $\begingroup$ Don't confuse the natural language with the mathematical language. $\endgroup$
    – Asaf Karagila
    Mar 12, 2015 at 14:30
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    $\begingroup$ While the answers emphasizing the technical definition are technically correct, I'd suggest that it's also intuitive, at least if you understand have the proper perspective on the empty set. Here, I'll count all the elements of the empty set: There we go, I did it. $\endgroup$
    – aes
    Mar 13, 2015 at 4:29
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    $\begingroup$ I think the other end of the scale is even less in line with the non-mathematical interpretation of the word "countable": to "count" some things in common English implies that at some point you have to have counted the last thing, so how can anyone possibly count all the natural numbers in that sense? Yet we say $\mathbb N$ is countable. ... In other words, "countable" has a specific technical definition in math, and you have to parse that definition rather than using something from casual use in daily life. $\endgroup$
    – David K
    Mar 13, 2015 at 12:25

5 Answers 5

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The empty set is a subset of $\mathbb{N},$ therefore a countable set.

For motivation, the intersection of two countable sets is a countable set, and the intersection of any two countable disjoint sets is the empty set.

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    $\begingroup$ OK thankyou but do we need to construct a function explicitly for a 1-1 corresponding relationship in that case I don't find any $\endgroup$
    – Uncool
    Mar 12, 2015 at 14:29
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    $\begingroup$ It takes some thinking to understand. The identity function from the empty set to itself is a 1-1 correspondence. Consider all the definitions involved carefully. See if you can convince yourself of it. $\endgroup$
    – GEdgar
    Mar 12, 2015 at 14:41
  • $\begingroup$ @GEdgar, so what is the identity function from the empty set to itself? When I think through all the definitions, I get $\{\{\{\}\}\}$. $\endgroup$ Mar 12, 2015 at 14:51
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    $\begingroup$ Let's say a function is a set of ordered pairs satisfying certain conditions. In this case, it happens to be the empty set of ordered pairs, $\{\}$. But Uncool will have to use the defintion of "function" in his own textbook. $\endgroup$
    – GEdgar
    Mar 12, 2015 at 14:55
  • $\begingroup$ @GEdgar, here's my thinking: An ordered pair is a set $\{a,\{a,b\}\}$, where $a\in A$ and $b\in B$. But if there is no $a$, the ordered pair is $\{\{b\}\}$, and if there is no $b$, the ordered pair is $\{\{\}\}$. So I get a function which is a non-empty set of ordered pairs. $\endgroup$ Mar 12, 2015 at 15:08
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In mathematics, as in all other technical fields from physics to furnace repair, there is a technical jargon in which ordinary terms, like “countable”, are used in particular technical ways that only somewhat resemble the ways the same words are used in ordinary speech. It is useful to think of the ordinary meaning of these words, as an aid to understanding them, but that ordinary meaning only gets you so far, and not as far as you need to get.

To understand “countable”, you can think of ordinary counting, but that will not get you all the way to understanding the mathematical meaning of “countable”, because the mathematical meaning is not the same as the ordinary meaning. In mathematics, “countable” has a particular technical meaning that you cannot guess from your ordinary understanding of this word. To understand the mathematical meaning, you must look up the mathematical definition.

The mathematical meaning of “countable” is:

A set $S$ is countable if there exists a subset of the natural numbers, say $T$, and a one-to-one correspondence $f$ between $S$ and $T$.

To understand this definition, you should look at how it corresponds to the ordinary notion of “countable”, and also how it does not. For example, the definition implies that the set of even numbers is “countable” in this technical sense, whereas the ordinary meaning of “countable” probably does not apply to the set of all even numbers, which has no beginning and no end. Similarly, the definition implies that the empty set is “countable” in this technical sense, although you have observed that to call the empty set “countable” in the ordinary sense is strange. But this just shows that the two senses are different, and you cannot use the ordinary sense as a replacement for the technical sense.

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  • $\begingroup$ Thankyou so much MJD now i,'be got it $\endgroup$
    – Uncool
    Mar 12, 2015 at 14:38
  • $\begingroup$ I'm glad I could help. $\endgroup$
    – MJD
    Mar 12, 2015 at 14:43
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HINT:

$\emptyset$ is countable if and only if there exists injective $f: \emptyset \to \mathbb{N}$

Let $f=\emptyset\times\mathbb{N}$. Yes, it really works!

First, prove $f$ is a function mapping $\emptyset \to \mathbb{N}$. Then prove $f$ is injective. Mostly, things will be vacuously true.

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A set is said to be countable if it is either finite or countable infinite.

The empty set is finite! (it is clearly intuitive)

The definition of finite sets says that a set is finite if there is a bijective correspondence of the set with some section of the positive integers. That is, the set $A$ is finite if it is empty or if there is a bijection $f:A \to \lbrace 1,2,3,\dots ,n\rbrace$ for some positive integer $n$.

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Though many of above answers are so Technically true. I think the best way to think of null set as a countable set is to remember that the empty set is a subset of any well defined set. Since the empty set is a subset of the Natural Numbers and the natural numbers is countable therefore the empty set is countable.

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