The Power of Two Sets theorem

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Just wanted to know if the work is sound and logical on my paper posted above.

I realized I probably should have included notation for the power of the sets. This is my first attempt at theorem proving in Introductory Real Analysis. I realize now that I’m starting into a subject that requires a great deal of creativity in mathematical thought that is not just mindless problem solving. My highest mathematics educations stopped at linear algebra after diff EQ. I have a bachelors degree in Mechanical Engineering.
 

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  • #2
tnich
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View attachment 230537 Just wanted to know if the work is sound and logical on my paper posted above.

I realized I probably should have included notation for the power of the sets. This is my first attempt at theorem proving in Introductory Real Analysis. I realize now that I’m starting into a subject that requires a great deal of creativity in mathematical thought that is not just mindless problem solving. My highest mathematics educations stopped at linear algebra after diff EQ. I have a bachelors degree in Mechanical Engineering.
I am not sure what you are trying to prove. The power of a set is not something I remember from Real Analysis, nor can I find it in any of my text books. Do you mean the order of a set?
 
  • #3
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The book i have is kind of vague with it but it seems they explain the power of a set as the number of elements in a set. At least it explains that the definition can be reduced to that.

I try not look to into the book much since the proof is right next to it, but they conclude that there is no set with largest power. There are sets with powers larger than other sets, but there is infinite amount of sets that could be of larger power from my interpretation. They use a contradiction between relations of sets. I tried to go about it on my own using my imagination and brain but I’m nowhere near as elegant as their proof even thought their proof is extremely wordy.
 
  • #4
tnich
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The book i have is kind of vague with it but it seems they explain the power of a set as the number of elements in a set. At least it explains that the definition can be reduced to that.

I try not look to into the book much since the proof is right next to it, but they conclude that there is no set with largest power. There are sets with powers larger than other sets, but there is infinite amount of sets that could be of larger power from my interpretation. They use a contradiction between relations of sets. I tried to go about it on my own using my imagination and brain but I’m nowhere near as elegant as their proof even thought their proof is extremely wordy.
What you are describing - the number of elements in a set - is called the cardinality of the set. You are also using the concept of the powerset of a set S, which is the set of all subsets of S. It sounds as though you are trying to prove that the cardinality of the powerset of M is greater than the cardinality of M.
 
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  • #5
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What you are describing - the number of elements in a set - is called the cardinality of the set. You are also using the concept of the powerset of a set S, which is the set of all subsets of S. It sounds as though you are trying to prove that the cardinality of the powerset of M is greater than the cardinality of M.
I see. This book was written in 1970s so it’s a little outdated. I should probably get an updated modern textbook. Got any recommendations?
 
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They also have a approach where they ascertain singular elements one to one corresponding to the subsets of A, B, X. They use that approach and I don’t use that approach period. Which is where i probably go wrong. My attempt was to prove that the power of curly M is greater than regular m without incorporating a one to one correspondence of the elements to their designated subset. I used a one to one correspondence between the subsets. Their proof is extremely wordy. So I really tried to tackle it with a more symbolic approach. I failed on it more than likely from my own misunderstandings.
 
  • #7
tnich
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I see. This book was written in 1970s so it’s a little outdated. I should probably get an updated modern textbook. Got any recommendations?
Real analysis has not changed much in the last 50 years, so a newer text would not be a great advantage. Long ago when I took it, Royden was held up as the standard for real analysis texts.
 
  • #8
tnich
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They also have a approach where they ascertain singular elements one to one corresponding to the subsets of A, B, X. They use that approach and I don’t use that approach period. Which is where i probably go wrong. My attempt was to prove that the power of curly M is greater than regular m without incorporating a one to one correspondence of the elements to their designated subset. I used a one to one correspondence between the subsets. Their proof is extremely wordy. So I really tried to tackle it with a more symbolic approach. I failed on it more than likely from my own misunderstandings.
If cardinality of ##\mathbb M## relative to ##M## is what you after, then the theorem is not true in general. A counter example is the empty set, which has only one subset - itself. Another is the set of positive integers, which has countably infinite cardinality. It is easy to enumerate all of its subsets, so the cardinality of its powerset is also countably infinite. So I think the first thing you need to do is understand what the author means by "power of a set".
 
  • #9
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@zeronem, it's not clear to me what you're trying to do. If the cardinality of a set S is n, then the cardinality of the powerset of S is ##2^n##. See https://en.wikipedia.org/wiki/Power_set.

Also, I agree with @tnich that just because an analysis book is about 50 years old, it doesn't mean that it is outdated.
 
  • #10
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This is about the only time in this whole book that they even use the word cardinal number which is at the very bottom, that which is 7 pages ahead of the theorem I made an attempt on lol. This book was translated from a Russian author. Maybe the approach at teaching this subject is a bit different from modern approaches.
 

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  • #12
mathwonk
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that discussion seems a little imporecise and colloquial to me, although not too bad. the key definiton of "equivalent" sets is omitted in that section but is the most important part of the notion of "same power". i.e. if you are careful you will notice they have not defined power of a set at all, rather they have defined when two sets have the same power. I agree that a 50 year old analysis book is not necessarily outdated, and in general i rather like russian books, but this one may not be as helpful to you as another one might be, maybe royden, or anything by george simmons or sterling berberian, i am curious who is the russian author of your book.

in your case i am guessing that the word "power" instead of order or cardinality, was just a strange choice of translation made by the translator. so i assume that translator was not an english speaking mathematician. that will make your book read oddly.
 
  • #13
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that discussion seems a little imporecise and colloquial to me, although not too bad. the key definiton of "equivalent" sets is omitted in that section but is the most important part of the notion of "same power". i.e. if you are careful you will notice they have not defined power of a set at all, rather they have defined when two sets have the same power. I agree that a 50 year old analysis book is not necessarily outdated, and in general i rather like russian books, but this one may not be as helpful to you as another one might be, maybe royden, or anything by george simmons or sterling berberian, i am curious who is the russian author of your book.

in your case i am guessing that the word "power" instead of order or cardinality, was just a strange choice of translation made by the translator. so i assume that translator was not an english speaking mathematician. that will make your book read oddly.

A.N. Kolmogorov and S.V. Fomin

Translated by Richard A. Silverman

I’ll most certainly look into the authors you listed. I appreciate it!
 
  • #14
mathwonk
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well that blows me away, because those are great russian authors and the translator is an english speaking mathematician who i recall wrote a superb book on complex analysis.. i am surprized.
 
  • #15
mathwonk
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ok i take back my reservations. that book looks indeed excellent and the only strange thing i see there is the use of the word "power" instead of "cardinality". they do not use the term "power set" for set of all subsets, so there is no confusion there. i think you can learn a lot from that book, but maybe berberian is easier to read.
 
  • #16
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It is a 24 dollar Dover publicated book. The Dover books tend to run through things pretty quickly. But the first Dover publicated book I ever got into authored by Max Born inspired and potentiated my advancement in education in mathematics and physics when I was in high school. So I’m big fan of these books! Lol
 
  • #17
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Here is their proof of the theorem. It’s really wordy. I tried to make my failed proof much more symbolic. I didn’t use any one to one correspondence between the elements in their subsets like they did in this proof.

I really try to take a theorem and visualize a structure maybe made up by Venn diagrams and work off of that structure with the concepts in mind to gather a proof that is more visual to me. Hopefully that is right mind set to tackle these theorems.
 

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  • #18
verty
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Well it was always going to be a diagonal argument, right? I mean, this is what I expected when I heard about it.
 
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  • #19
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The book i have is kind of vague with it but it seems they explain the power of a set as the number of elements in a set. At least it explains that the definition can be reduced to that.

I try not look to into the book much since the proof is right next to it, but they conclude that there is no set with largest power. There are sets with powers larger than other sets, but there is infinite amount of sets that could be of larger power from my interpretation. They use a contradiction between relations of sets. I tried to go about it on my own using my imagination and brain but I’m nowhere near as elegant as their proof even thought their proof is extremely wordy.
It looks like you are dealing with basic set theory here. A good, very readable, and fascinating exposition is Naive Set Theory by Halmos. Enjoy. I did; I found it mind-stretching. It's another old book, by the way.
 
  • #20
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It looks like you are dealing with basic set theory here. A good, very readable, and fascinating exposition is Naive Set Theory by Halmos. Enjoy. I did; I found it mind-stretching. It's another old book, by the way.
Thanks! I will look into it!
 
  • #23
mathwonk
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well it's worth it.
 
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