Talk:Cardinality

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Contents 1 Database theory 2 Cardinality of the powerset of the natural numbers 3 intro wording 4 Comment

[edit] Database theory

To include treatment about cardinality as it relates to database theory (i.e., that cardinality refers to the relationships from one entity to another or that it represents the number of tuples in a table), would that be sensible? Or is it deserving of a separate article?

To Lazyboyz: 1. Databases are not part of set theory. Your suggested contribution should be in a different article in a different category. I suggest that you try to find something in the area of computer science for it. 2. This section of the discussion which you created should have been put at the end, not the beginning of this page. Also you should not creat section headers manually as you did. You should use the "+" tab at the top of the screen to initiate a new section of the discussion. It will prompt you for a section title and put the new section at the end as is proper. 3. You should sign your contributions to the discussion by putting four tildas, like "~~~~", at the end of your message. The software will automatically replace it with your user-id and date and time when you save the addition. As I now do here: JRSpriggs 07:26, 25 April 2006 (UTC)

JRSpriggs, i'm not lazyboyz, but I wanted to comment on something. i always creat section headers, then again I always include new comments on the bottom of the page. I didn't know you HAVE to use the "+" tab, which by the way, appears nowhere on top of the screen. Let me know where it appears so I can use it next time, and please tell me where it says I HAVE to use that tap instead of manually creating a section title. regards,Cjrs 79 14:27, 25 April 2006 (UTC)

To Cjrs 79: I said that Lazyboyz "should" use the plus tab, not that he or anyone-else "must" use it. Since he seems to not understand our system well, then I thought that using that method would help him do the right thing instead of making mistakes. The plus tab is at the top of discussion (talk) pages between the "edit this page" tab and the "history" tab. By the way, your user name includes my initials ("jrs"). So I am curious as to what it stands for. JRSpriggs 10:26, 30 April 2006 (UTC)

[edit] Cardinality of the powerset of the natural numbers

The assertion that 2^Aleph-null is the cardinality of the real numbers is troubling me. Isn't it possible to enumerate the power set of the natural numbers? It seems so to me, I started doing it. Say that the empty set is the first element (that is, we're creating a function mapping the naturals to their power set, so f(0) would be the empty set), f(1) would be {1}... f(2) = {2} f(3) = {1, 2}

f(4) = {3} f(5) = {1, 3} f(6) = {2, 3} f(7) = {1, 2, 3}

f(8) = {4} f(9) = {1, 4} f(10) = {2, 4}...

and so on (forever). Doesn't creating this mapping show that the two sets are both countable? Or is my mapping somehow not good enough?

Thanks, -David

Well David. Any function between reals and naturals is never surjective. What you assert in your comment is that the naturals are enumerable, or countable, which can be proven false by a diagonalization argument. Also note that you would need a 1-1 correspondece. Can you prove that the function you just defined is indeed a 1-p1 correspondence?

I hope this helps, if not feel free to leave a message in my user page.

Cjrs 79 18:23, Jan 17, 2005 (UTC)

The problem with your enumeration, David, is that you will only get finite subsets that way, not infinite subsets. -- Walt Pohl 22:39, 17 Jan 2005 (UTC)

Thanks for your answers. I don't quite understand yet how my enumeration is invalid, but I'm mulling over the idea that it doesn't generate infinite sets. The problem I am really wrestling with is this: Why does the power operation create a strictly higher cardinality, as opposed to other operations (addition, multiplication). I've read Cantor's proof, and seeing no flaws I guess I accept it, but since it works through setting up a contradiction, it doesn't really speak to my question, as far as I can see. All it tells me is that 2^Aleph-n > Aleph-n. (To say that 2^Aleph-n = Aleph-(n+1) seems like an unfounded extrapolation, unless there's a proof of this I haven't seen.) For what values of k would k^Alpeh-n > Alpeh-n, anything larger than 2? Larger than 1? How could I find out generally what functions f( k, Aleph-n ) > Aleph-n. Or what about Aleph-n * Aleph-n. To me this would intuitively be Aleph-(n+1), but we already know that I don't have too strong a grasp on all this business. :P

[edit] intro wording

The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number.

I dislike this wording. It's perfectly possible to discuss cardinality without reference at all to cardinal numbers, you simply say two sets have the same cardinality if there is a bijection between them. In fact, this was the approach taken by Frege, I think. I'm not arguing against cardinal numbers by any mean, simply suggesting that the notion of cardinal number is not necessary to discuss cardinality, so the intro should be reworded. Revolver 09:45, 14 Jun 2005 (UTC)

Wouldn't that be like saying "two groups are isomorphic if there is a.... , instead of an isomorphism is a...." I think there should be a discussion or explanation of what cardinality means outside of the property of two set having the same cardinality. Maybe we can find a definition that we can agree on...

Cjrs 79 12:58, Jun 14, 2005 (UTC)

I'm not sure what your comment means. You can talk about cardinality simply by defining bijection, just as you can define isomorphic groups simply by defining group isomorphisms. The difference is when you select a representative from each isomorphism class, in the case of cardinality these are the cardinal numbers. To do the same for groups would be to select a group representative from each group-isomorphism class, and then collect these into a class itself. (More precisely, you would be forming the skeleton of Grp. I'm not sure what "I think there should be a discussion or explanation of what cardinality means outside of the property of two set having the same cardinality." means. By definition, "two sets have the same cardinality" if there is a bijection between them. That is the definition of the term "cardinality". Then, you can define cardinal numbers and prove two sets have the same cardinality iff they have equal cardinal numbers. Revolver 19:48, 14 Jun 2005 (UTC)

[edit] Comment

This is a different question from the above, but since I don't feel like starting a new topic, I'll post it here:

I read somewhere that the reals in the interval [0,1] can be put into a bijection with the \Re \geq 0 . The bijective function in question is \frac{x}{1-x}

How can it be proven that the function is bijective?

I (naively) thought of the following:

Suppose there exist a nonnegative real number r. Then \frac{x}{1-x} = r which can be expressed in terms of r to be x = \frac{r}{1+r}

Now it must be shown that 0 \leq \frac{r}{1+r} \leq 1

Suppose \frac{r}{1+r} < 0Then r < 0 which contradicts the definition of r. Suppose \frac{r}{1+r} > 1, then r > 1+r, which means 1 < 0, which is a contradiction. The function can be shown to be injective by noting that its derivative \frac{1}{(1-x)^2} is always positive for all values of x and hence the function is strictly increasing and hence injective.

But the limitations of my informal "proof" are that it does not show that the function is surjective. How can this be shown? If there is anything with the "proof" above, please point it out; I haven't studied maths formally yet. —The preceding unsigned comment was added by 129.170.67.78 (talk • contribs).