Talk:Double counting (proof technique)

message: The article begins as follows:
:In combinatorics, double counting, also called two-way counting, is a proof technique that involves counting the size of a set in two ways in order to show that the two resulting  expressions for the size of the set are equal. Such a proof is sometimes called a bijective proof, when it depends on showing the existence of a bijective mapping. That is, we may consider what we are doing to be taking a finite set X and counting it by method A and then method B; or we may also take two sets X and Y and count them both, then show by means of a bijection f from X to Y that the results are the same. The abstract difference in presentation can usually be absorbed in the freedom to count in various ways.
It seems to me that this confuses two different things with each other.  (1) In bijective proofs one finds a bijection between two different sets.  (2) In double counting, one counts the members of just one set in two different ways.  If these are really both the same thing, that should be explained. Michael Hardy 23:27, 11 Mar 2005 (UTC)
thread_title: Untitled

message: Actually, there are two pricinples involved under the same name.
I will explain them detail in the next days when I find some spare time.
# Bijective proof. One has to find bijection (one to one and onto mapping f:A -> B. This proves that |A| = |B|.
# Bookkeper's rule. Take a matrix M. The sum of elements of M can be computed in two ways. 
1. Take the sum of each row and then the sum of all row-sums.
2. Take the sum of each columns and then take the sum of all column-sums. Tomo 23:47, 11 Mar 2005 (UTC)
thread_title:  Another opinion 

message: I suggest that the mathematics discussion be separated from the accounting discussion.    -- Dominus 02:38, 10 April 2006 (UTC)
replies: text: I have moved the accounting discussion to double counting (accounting). -- Dominus 18:46, 24 April 2006 (UTC)

text: Note that I did not create Double counting (proof technique) back in April; I did what I described above, and moved the accounting stuff to Double counting (accounting) while leaving the mathematical discussion at Double counting (with no qualifier).   -- Dominus 14:07, 3 August 2006 (UTC)
replies: text: I'm the one who added the qualifier; a complete description of the movements appears below. Stebulus 18:52, 3 August 2006 (UTC)

text: Oops.  I totally missed your comments here, JA.  Earlier today I turned Double counting into a disambiguation page, moved its prior content to Double counting (proof technique), split off the section on the fallacy to Double counting (fallacy), and changed Double-counting into a redirect to Double counting.  (The hyphenated version was originally about accounting, which is in Double counting (accounting).)  All this seemed like the obvious thing to do... I now see that I should have checked here for prior conversation on the topic.  Well.  Now that we're here, any comments or objections?  Stebulus 22:46, 2 August 2006 (UTC)

text: ...Fast forward to late 2008 and there's still no explanation about non-trivial bijections from the set to itself. I'm boldly removing that paragraph. Melchoir (talk) 08:59, 18 October 2008 (UTC)
thread_title: Proposed split