Difference between least, minimal element

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The discussion clarifies the difference between a least element and a minimal element in set theory, noting that a least element is smaller than all others, while a minimal element is not larger than any other element. It explains that a poset with a least element has a unique least element, but can have multiple minimal elements if it lacks a least element. Participants also share insights on learning set theory and suggest resources for understanding proofs, emphasizing that proofs should be understood rather than memorized. Recommended books include "How to prove it: A structured approach" by Velleman and "Naive Set Theory" by Halmos, although the latter is not ideal for beginners. Overall, the conversation highlights the complexities of set theory and the importance of grasping foundational concepts.
kadas
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Can you guys explain to me what is the difference between least element and minimal element? I keep struggling to understand the difference between them but till now i still cannot resolved it.

right now, i am trying to learn set theory, do you have any reference(book, link,whatever) that i can refer to while i am stuck with set theory?

i also noticed that there are a lot of proving in the book when they try to build the foundation of set theory, do you have any guidelines on how to do a proving?
 
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A least element is an element smaller than all other elements. I.e. x is least if for all y we have,
x \leq y
A minimal element is one that is not larger than any other element. I.e. x is minimal if for all y, either x and y are incomparable or x \leq y.

If a poset has a least element, then it's unique and the poset cannot have any other minimal elements (because then the least element would be smaller and the minimal element wouldn't be minimal anyway). However if a poset does not have any least element, then it may have many minimal element. A straightforward example is to consider all pairs (a,b) of non-negative integers and order them by (a,b) < (c,d) if and only if a<c and b<d. Then all elements on the x-axis and y-axis are minimal (i.e. elements of the form (a,0) or (0,b) are minimal).

right now, i am trying to learn set theory, do you have any reference(book, link,whatever) that i can refer to while i am stuck with set theory?
Why not just refer to whatever you're learning from?

i also noticed that there are a lot of proving in the book when they try to build the foundation of set theory, do you have any guidelines on how to do a proving?
A proof is not something you remember, but something you understand. In many other subjects you can just remember the contents, but when trying to do proofs this is not the correct approach. A proof is simply an exposition of your thoughts on why a certain statement is true and the hard part is getting used to actively thinking.

Through observing other people's proofs and doing a lot of your own you should get better at it. I have heard good things about "How to prove it: A structured approach" by Velleman, but haven't read it myself. "How to solve it" by Polya is a classic on mathematical problem solving which I like myself, but it doesn't focus on proofs as such, just problem-solving. This means that it doesn't describe propositional logic, try to make you remember various arguments Latin name, etc. In my opinion this is a good thing since it gets down to the essentials, but if you're very inexperienced you may need a bit more guidance.
 
wow! thank you for your answer, that's enlightening for me...by the way, i am a physics student, it just happen that I have to learn set theory and I am not used to the way mathematician describe everything in a very refined way..I used to think that " this one or that one is "quite obvious"", but it is only after i started to learn pure mathematics and i realized that it is not very very very obvious to write down the proof..hahha..

Anyway, may I know what is your reference book you used when you study set theory?
 
(Assuming you're replying to me. There was another reply shortly before mine, but it seems to have been deleted. If you meant for that poster to get the reply disregard this post.)

kadas said:
Anyway, may I know what is your reference book you used when you study set theory?
I mainly learned the very basics of set theory from the various introductions that many introductory math books start out with (or have as an appendix). Later on when I was comfortable using the language of set theory, but wanted a good understanding of it I picked up Naive Set Theory by Halmos which is a great book, but not really good as a first exposure to set theory. I haven't really come across a good exposition of set theory for the complete beginner.
 
hmm...thanks for the reference, i'll try to take a look at it.hahha..
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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