I’m sitting here thinking of things to write for my first blog and coming up with nothing. I suppose the main purpose of this SLOG is to create discussion about previous or future problems/proof in the class so that’s probably a good place to start.
So far we have covered simple and complete induction and the principle of well ordering. The principle of well ordering, to me, seems like a formal way of saying something that is inherently true about Natural Number based on their construction; it seems no more relevant to me than, when using an array for some program, to say a nonempty array has a first and last element. But having said that, I understand the need to formalize mathematical proofs and this principle provides a means to do that.
As far as simple and complete induction, I’ve seen these types of proofs in several mathematics and computer science courses I’ve taken and it seems that some of these proofs rely on reasoning inside a formal proof which seems to give the reasoning validity, but to me it seems only slightly stronger, if at all, than arguing the proof by reason without the use of induction.
For example P(n): Every set of n elements has exactly 2n subsets, the proof of which uses the fact that every subset of a set S can be considered as those that contain a particular element and those that don’t. Because of this, there is a 1-1 correspondence between the two subsets and thus as set of n+1 elements is composed to two sets of subset, those with element and those without, each of which has 2n elements (by induction hypothesis). I understand this to be true, however there doesn’t seem to be as much validity to a proof of this nature compared to one that employs algebra or one that can be easily verified by the reader, such as the proof of 12n – 1 is a multiple of 11, which I personally find to be logically and algebraically simple and easier to follow therefore making it a better proof.
I suppose the nature of mathematical problems doesn’t allow for every proof to be constructed in a clear concise manner that even people without a mathematical background could grasp. But it does seem that induction is the most useful and efficient tool for proving something for a large (or infinite) set of numbers that would otherwise not be so easily proven, with or without reasoning.
So far we have covered simple and complete induction and the principle of well ordering. The principle of well ordering, to me, seems like a formal way of saying something that is inherently true about Natural Number based on their construction; it seems no more relevant to me than, when using an array for some program, to say a nonempty array has a first and last element. But having said that, I understand the need to formalize mathematical proofs and this principle provides a means to do that.
As far as simple and complete induction, I’ve seen these types of proofs in several mathematics and computer science courses I’ve taken and it seems that some of these proofs rely on reasoning inside a formal proof which seems to give the reasoning validity, but to me it seems only slightly stronger, if at all, than arguing the proof by reason without the use of induction.
For example P(n): Every set of n elements has exactly 2n subsets, the proof of which uses the fact that every subset of a set S can be considered as those that contain a particular element and those that don’t. Because of this, there is a 1-1 correspondence between the two subsets and thus as set of n+1 elements is composed to two sets of subset, those with element and those without, each of which has 2n elements (by induction hypothesis). I understand this to be true, however there doesn’t seem to be as much validity to a proof of this nature compared to one that employs algebra or one that can be easily verified by the reader, such as the proof of 12n – 1 is a multiple of 11, which I personally find to be logically and algebraically simple and easier to follow therefore making it a better proof.
I suppose the nature of mathematical problems doesn’t allow for every proof to be constructed in a clear concise manner that even people without a mathematical background could grasp. But it does seem that induction is the most useful and efficient tool for proving something for a large (or infinite) set of numbers that would otherwise not be so easily proven, with or without reasoning.
2 comments:
You're right: induction is one of the best tools for dealing with large or infinite sets (provided they have some connection to the natural numbers).
Now, the fact that a proof is about a domain that hasn't been reduced to algebra or formulas doesn't make it less rigorous or precise. We can use English as our notation, and demand the same high standards as we demand of equations.
That is a good point. I suppose I didn't think of english as a notation for expressing some kind of mathematical idea.
Although I feel that a mathematical proof is, I guess, more concise using mathematical tools (algebra, calculus etc.) rather than english.
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