- #1
StatOnTheSide
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Hi all. I am currently studying Niave Set Theory by Halmos. I love that book. It has been going really well so far especially with the help of people form Physics Forums. I am also studying Linear Algebra from Axler's book but that is a cake walk comared to Set Theory simply because I am new to this level of abstraction. I am an engineer by profession and am not in a school studying math. I read it in the evening after work and have no guidance except for that provided by the wonderful people of this forum.
I have few questions regarding excercises.
1. Is it necessary that I should solve all the problems before I move on to the next chapter?
I usually end up not being able to solve some. The latest one has been "prove that any non empty subset E of a natural number has an element k such that it belongs to all the other elements in E which are not equal to it." I know that this is another way of wording the well ordering principle; however, Halmos expects me to solve well ordering principle before moving on to the next chapter. I don't know about others but for me, it is really hard to prove it.
2. How many people in math major actually solve ALL the problems that are present in Halmos's book?
I know that it should not matter as to how good others are but I wish to know it because it would be good to know the culture of the aspiring mathematicians.
3. How can I develop my ability to construct proofs?
I am really good at proving some kinds of propositions. There are others which I find are lot harder. I see that there are a lot of books on proof writing like "How To Prove It: A Structured Approach by Daniel J. Velleman", "The Nuts and Bolts of Proofs by Antonella Cupillari " and "How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow". I bought the first edition of the last book (Solow's book). I can see that most of the mathematical proofs have a structure to it. Understanding the srutcture will greatly help but will nevertheless not help me ALL the excercise problems in a set theory book.
4. Is there any way out of this?
I'd greatly appreciate it if you can kindly give your feedback to my questions that I have asked in the above paragraphs.
I have few questions regarding excercises.
1. Is it necessary that I should solve all the problems before I move on to the next chapter?
I usually end up not being able to solve some. The latest one has been "prove that any non empty subset E of a natural number has an element k such that it belongs to all the other elements in E which are not equal to it." I know that this is another way of wording the well ordering principle; however, Halmos expects me to solve well ordering principle before moving on to the next chapter. I don't know about others but for me, it is really hard to prove it.
2. How many people in math major actually solve ALL the problems that are present in Halmos's book?
I know that it should not matter as to how good others are but I wish to know it because it would be good to know the culture of the aspiring mathematicians.
3. How can I develop my ability to construct proofs?
I am really good at proving some kinds of propositions. There are others which I find are lot harder. I see that there are a lot of books on proof writing like "How To Prove It: A Structured Approach by Daniel J. Velleman", "The Nuts and Bolts of Proofs by Antonella Cupillari " and "How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow". I bought the first edition of the last book (Solow's book). I can see that most of the mathematical proofs have a structure to it. Understanding the srutcture will greatly help but will nevertheless not help me ALL the excercise problems in a set theory book.
4. Is there any way out of this?
I'd greatly appreciate it if you can kindly give your feedback to my questions that I have asked in the above paragraphs.
