Difference between least, minimal element

[kadas]

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?

 

[rasmhop]

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.

 

[kadas]

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?

 

[rasmhop]

(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.)

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.

 

[kadas]

hmm...thanks for the reference, i'll try to take a look at it.hahha..