Talk:Addition principle

Redirection

I redirected this page to combination because the idea that was expressed here (that the number of elements in the union of disjoint sets is the sum of the number of elements in each set) is so basic to arithmetic that it scarcely needs elaboration, and because the rest of this article dealt with the same subject matter as the existing "combination" article does. DavidCBryant 17:22, 15 April 2007 (UTC)

Here, too, DavidCBryant, you replaced a stub about one of the wide-spread notions in combinatorics with a redirect which does not address the subject of the former stub. I disagree with your arguments for this.

The addition principle to some extent, and the principle of multiplication or product rule or rule of product to a very high extent, are "self-evident truths once you understand them", but are treated as combinatorial cornerstones in elementary courses in combinatorics. The same goes e.g. for the pigeon hole principle; so why do textbooks in combinatorics (and we university level math teachers) spend so much time and space on these trivialities?

I personnaly have three answers to this. The first is, that, the principles are not "self-evident" until you have understood them. As teachers, our task ideally is to get the students to understand that the basic ideas in combinatorics are there for them to grasp, by their own clear thinking, not by memorising different formulae to be applied in different but very similar sounding situations. The "simple ideas" are made simple to understand, by bringing them out in the open. Attributing names to them is one way to pinpoint them.

Second, while these principles indeed are very trivial when you think about it, their applications often are very far from trivial. David, I'm sure you find it absolutely self-evident that if 101 or more pigeons enter 100 pigeon holes, then there is at least one hole into which at least two pigeons enter. However, do you find the following common beginner's application of the pigeon principle equally trivial:

If you choose 101 of the integers between 1 and 200, then there are two among them, such that one divides the other.

(Recall that the integer a divides the integer b, if and only if there is an integer c, such that ac = b.) You might choose 100 such integers without this property, namely e.g. 101, 102,..., 200. If you started by picking precisely these integers, of course you cannot pick another one which does not divide at least one of the ones you already have picked; but what excludes any other choice of 101 of the integers, too?

In order to solve this problem, you probably will have to combine several ideas. One of these is the extremely trivial pigeon hole principle. (Another idea is that a smaller power of two will divide a larger one; and a third is that every positive integer is the product of an odd number and a power of two.) However, without having made it explicit, you might not get the idea to apply this trivial idea in this context.

Similar applications in complex situations are common for the multiplication and addition principles, too.

Third, actually quite a lot of mathematic theory may be broken down to a very large amount of small "self evident" statements. Perhaps, this is more visible in combinatorics and in elementary arithmetics than in other fields of mathematics; but the phenomenon is ubiquitous, if you analyse the theories carefully enough. Mathematic theories could be compared to rather complicated consisting of very simple building blocks. It is not the building blocks, but the intricate ways they are combined, that are non-trivial. However, it is hard to explain the theories, until you have made their constituents visible.

Conclusion: I'm inclined to revert your redirects. Is this OK for you?-JoergenB (talk) 05:19, 15 December 2007 (UTC)