Difference between least, minimal element

In summary, the difference between a least element and a minimal element in a poset is that a least element is smaller than all other elements, while a minimal element is not larger than any other element. If a poset has a least element, it cannot have any other minimal elements. However, if it does not have a least element, it may have multiple minimal elements. As for learning set theory, it is recommended to refer to the material being studied and to actively think about the proofs rather than trying to memorize them. Some resources for learning how to do proofs include "How to prove it: A structured approach" by Velleman and "How to solve it" by Polya. A good reference book for set theory is
  • #1
kadas
12
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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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  • #2
A least element is an element smaller than all other elements. I.e. x is least if for all y we have,
[tex]x \leq y[/tex]
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 [itex]x \leq y[/itex].

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.
 
  • #3
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?
 
  • #4
(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.
 
  • #5
hmm...thanks for the reference, i'll try to take a look at it.hahha..
 

FAQ: Difference between least, minimal element

What is the difference between a least element and a minimal element?

A least element is the smallest element in a partially ordered set, meaning that it is smaller than or equal to all other elements. A minimal element is the smallest element in a subset of a partially ordered set, but may not be smaller than all other elements in the entire set. In other words, a minimal element is only guaranteed to be smaller than or equal to other elements in its subset, while a least element is guaranteed to be smaller than or equal to all elements in the entire set.

Can a partially ordered set have more than one least element?

No, a partially ordered set can only have one least element. This is because if there were two least elements, they would both have to be smaller than or equal to each other, making them equal. However, a partially ordered set can have multiple minimal elements if they are all in different subsets.

Is the least element always unique?

Yes, the least element is always unique in a partially ordered set. This is because if there were two least elements, they would both have to be smaller than or equal to each other, making them equal. Therefore, there can only be one least element in a partially ordered set.

How is a least element different from a minimum element?

A minimum element is the smallest element in a totally ordered set, meaning that it is smaller than all other elements. In contrast, a least element is the smallest element in a partially ordered set, which only requires it to be smaller than or equal to all other elements. Therefore, a minimum element is always a least element, but a least element may not always be a minimum element.

Can a partially ordered set have a least element and no minimal elements?

Yes, a partially ordered set can have a least element and no minimal elements. This can happen if the least element is the only element in the set or if all other elements are incomparable to the least element. In this case, the least element is the only element that is smaller than or equal to all other elements, but there are no elements smaller than or equal to it, making it the only least element and the only minimal element.

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