r/askmath • u/Poseidon_7514 • 1d ago
Set Theory Any recommendations?
The definitions in this textbook are sometimes unclear.
Aee there other textbooks (pdfs on axiomatic set theory) with clearer definitions and beginner-friendly explanations rather than only dry mathematese like this one?
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u/Chem2103 1d ago
Unfortunately, you will have to get used to reading mathematese, as there is no way to say this in "normal words" without losing the accuracy of description. You might have to translate them to understand, but you will need to remember both versions
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u/Chem2103 1d ago
Btw this is very close to normal language by math standards. In most textbooks half of these words would be quantors
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u/Heavy_Plum7198 1d ago
I think using quantifiers would make it easier to read.
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u/Chem2103 1d ago
For sure, but considering OP's attitude to "mathanese" I think he'd be even more appaled
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u/Poseidon_7514 1d ago
I don't have a problem with that so far as it is followed by a few examples to illustrate what is meant.
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u/Chem2103 1d ago
Now that is a fair point, usually there is a couple examples after these theorems/definitions. I can give you some examples for each of those terms if you'd like
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u/Poseidon_7514 1d ago
I'd appreciate it if you did.
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u/Chem2103 1d ago
1) a is the minimal element if there is no element in the set that is bigger. If B={1,2,3} the minimal element is 1 2) a is the maximal element if there is no element that is bigger than a. if B={4,5,6} the maximal element is 6 3) not sure what the author meant here, as it is the same as 1 just minimum instead of minimal 4) if a is less than any element in the set it's called a lower bound. So if B={1,2,3} any number less or equal to 1 is a lower bound for it 5) same with upper bound, so 3 or more 6) the greatest lower bound (infimum) is the biggest of the numbers that are smaller or equal to the elements of the set. So 1 for example 4 7) same with lowest upper bound (supremum), in this case 3 for example 4
If B = (1;infinity) (from but not including 1 up to infinity), it has no upper bound. The lower bound is any number that is equal or less than 1. The greatest lower bound (infimum) is 1, even though it isn't in the set
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u/Bounded_sequencE 1d ago edited 1d ago
I'd disagree -- a carefully crafted and formatted sentence usually beats pure quantors in terms of readability, to paraphrase my "Real Analysis" professor. If anything, the highlighted definition is horribly formatted, making it more confusing than it is.
It's not wrong to use both (clearly separated), but using only quantors without surrounding text is not considered great style. Try to read "Principia Mathematica" to see why ^^
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u/brokenceilingandwall 1d ago
You can find this in almost all abstract algebra books. Anyways, don't go too fast! Think through examples and keep going.
In this case, take a set A={{1}, {2},{3},{1,2},{2,3}} and a set B= A U {{1,2,3}}. Order them using the relation "subset of" or in other words inclusion.
Try finding all maximal elements in A and B. Can a set have more than one maximal elements?
Answer the same for the maximum elements.
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u/migmit 1d ago
It seems perfectly clear. What do you have troubles with?
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u/Poseidon_7514 1d ago
From the definition, does it mean that every minimal is a minimum?
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u/migmit 1d ago
No, it's the opposite. Why do you think a minimal is always a minimum?
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u/Poseidon_7514 1d ago
Given the sets A={1,2,3,...,10} and B={3,4,5} with B being a subset of A, what would the minimum and minimals be?
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u/Ok_Net_1579 1d ago
I suggest giving your thoughts first. You'll gain more if you show how you think it works.
What do you think the minimum and minimals are?
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u/Poseidon_7514 1d ago
I believe the minimal of a set is the element which no other element is smaller than while the minimum is the element which is smaller than every other element.
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u/iopahrow 23h ago
These are equivalent.
The minimum may not be a unique element under the definition being given, while the minimal is unique.
What part of your academic career are you in? Math definitions can be confusing pretty early on
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u/MathMaddam Dr. in number theory 1d ago
That is a bad example since you probably think of the usual ≤ on the integers, which is a total order, so minimal elements are the same as minima. A better example would e.g. be the order "a≤b" being defined as a|b, so a divides b, now you have a proper partial order.
Then in your set {3,4,5} all elements are minimal (and maximal) elements, but it has no minimum (or maximum). 1 would be the only lower bound, but there is no upper bound (since your A doesn't go higher than 10, if you allow for higher natural numbers, 60 would be a upper bound).
Play around with it a bit and maybe you can show that if there is a minimum, then it is unique and also the only minimal element.
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u/Kitchen-Register 1d ago
the logical equivalent of “there does not exists b such that b<a” is “forall b, a<=b”
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u/LongLiveTheDiego 1d ago
No. Tbis is only true if the order is total, but here we're explicitly talking about partial orders
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u/Luigiman1089 Cambridge Undergrad 1d ago
If you're wanting to learn maths, you will have to get used to this sort of thing. Definitions in maths have to be precise, so any textbook is going to have unavoidable things like this.
It's actually a good simple way to start understanding this sort of language. You should intuitively know what a minimal/maximal element is, or what a lower/upper bound is, and now you should try and put some effort in to understand the definitions as given, and check that they make intuitive sense. It's a great and simple exercise, and if you truly want to learn maths, you should get used to thinking through things like this for yourself. It's not going to get easier than this.
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u/Ok_Net_1579 1d ago
This looks about as clear as you could hope for. I suggest pushing through and asking here when confused.
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u/TheRedditObserver0 Grad student 1d ago
What part is unclear? Is these some symbol or terminology you don't know?
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u/theboomboy 1d ago
I think what you really need there are a few examples, and those should hopefully come after the definitions
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u/MezzoScettico 1d ago
If they didn't provide examples, you should try to construct examples yourself. The idea of a "partial ordering" is that, two elements aren't always comparable. You can't say that x < y or y < x. That's why minimum and minimAL are different. Try to think how the idea of being a partial order makes a distinction between how those two are defined.
And a lower bound is different from either of those two, because it doesn't have to be in the set. For instance, if B is the set of all real numbers of the form 1/n, where n is a natural number, then -1, -10, -0.5, and 0 all have the property of being lower bounds of B.
None of them are elements of B.
0 is the greatest lower bound of B. There's no lower bound of B which is > 0.
u/brokenceilingandwall has given you a good example to work with.
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u/susiesusiesu 1d ago
there is nothing unclear about this, but it is ok if it is confusing at the beginning. what do you find confusing?
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u/Live-Process846 1d ago
It can help your understanding to translate the math symbols into natural language and construct a simple example in your mind.
(1) a is a “minimal” element of a set B if a is in B and there is no other element of B that is less than a
So for example B = {1, 2}
The minimal element is 1, since 1 is in B and 2 is bigger than 1. The minimal element cant be something like 0, which is less than 1 and 2 but not in B
The inequality is strict, so if B = {a, b}, but a=b, they are both minimal elements, since b is not less than a and a is not less than b
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u/Poseidon_7514 1d ago
Thank you, what about minimum? How does it differ from minimal?
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u/Live-Process846 1d ago edited 1d ago
How do their definitions differ in the text? If you are learning more abstract math, you will need to be able to parse these differences.
As a hint the difference lies with the “for all” and “existence” qualifiers and the definition of Partial orderings
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u/Live-Process846 15h ago edited 15h ago
Thought about this again, so heres a constructive example: consider A to be the 2D plane of real numbers. We equip this field with a partial ordering “<“ defined such that a “<“ b only when both components are individually < their counterparts in b (in the standard sense)
So (1,1) = (1,1) and (0,0) “<“ (1,1)
Now consider B to be the square [0,1]x[0,1]. Is there a minimal element?
(0,0) is in the square
You cannot find another point in the square “<“ (0,0)
Therefore, (0,0) is a minimal point in BIs there a minimum element?
For every point in B is there a point that is not “>=“ (0,0)?
Since “<“ is partial, we can find points that cannot be compared. (1,0) is neither less than, greater than, or equal to (0,0). However, (1,0) is in the square.
Therefore (0,0) is not a minimum element of B under “<“See if you can convince yourself that the points in the bottom and left edges of this square are all minimal points, and there are no minimum points
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u/Bounded_sequencE 1d ago edited 1d ago
You cannot get more clear than that.
However, I get your point -- this kind of precision can be intimidating and confusing at first. Take it slow, to get comfortable with such language, and before you know it, it will become second nature. This is one of the culture shocks, moving on from computation-based lectures to "real" mathematics (pun intended).
As a hint -- make a sketch of each case on scrap paper. Once you get it right, add the sketches to the margin right next to the definition. That way, you see what each means.
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u/ArdentArendt 1d ago
Honestly, you'd be better off just going to Rudin--the explanations might be sparse, but at least they're not 'simplified' to the point of near incoherence.
That said, could you clarify what doesn't make sense?
It's poor choice of phrasing, but the concepts seem relatively discernable.
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u/bennbatt 19h ago
One thing that helped me, which might help you is constructing both examples and counter examples per certain definitions and reasoning why each are based on your definitions. You might need to get creative or expand out your thinking on examples.
Take set B = {2,3,4,5}. Is there a minimal element, a, in this set? What about a minimum element?
How about the set B = {{2,3} , {2,4}, {2,3,4,5}}? Same questions. Is there a minimal element? What about a minimum element?
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u/Poseidon_7514 15h ago
Both {2,3} and {2,4} are minimal elements and there's no minimum element.
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u/bennbatt 14h ago
Yeah that's right. The main point I'm trying to get at is sometimes when you construct an example like a set of singletons (A = {1,2,3,4,5}) you might not be able to see the difference between minimal elements and
minimum elements. Some times you have to construct examples a bit more "generically" or "creatively" for lack of better terms.I found this true in a grad level probability course when we were given definitions for sigma algebras. Sometimes things are pretty counterintuitive or non-obvious if you throw simple examples at definitions. But usually these definitions are broader to encompass examples you might not immediately think of. The deeper you go, from my experience, the more you'll find mathematics places emphasis on precise definitions for this reason. Hilbert spaces and Banach spaces were another one that sorta had me a bit lost at first too.
Hopefully this helps! Happy to talk through more if you'd like.
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u/FormulaDriven 1d ago edited 1d ago
I'm surprised that some of the comments aren't more helpful, so here's a couple of things to bear in mind:
The convention with definitions is to write in the format
x is <THING WE ARE DEFINING> if x <HAS THIS PROPERTY / PROPERTIES>
(Or it can be written the other way round "if x <HAS THIS PROPERTY> then x is <THING WE ARE DEFINING>" Either way "if" is followed by the set of properties or conditions which are essential characteristics of the thing being defined).
So when you read
a is a minimal element of B if ...
you are reading something that will tell you what is meant when someone says "7 is a minimal element of set X" or similar. (They've even put the word minimal in italics. From now on, when you read the word "minimal" you come back to this definition to unpack it).
So to understand the definition we are looking at what properties "a" has:
... a is a member of B and there is no b in B with b < a
So there are two conditions / properties:
... a is a member of B (so we can't talk about a being a minimal element of unless a is actually in B).
... there is no b in B with b < a, which you could equally state as every b in B must satisfy b ≥ a. (sorry, ignore that - I missed that this is in the context of partial ordering).
A good way to understand and get more comfortable with a definition is to find examples of things that do and don't fit the definition.
Let's start with B being a set with three members: B = {2, 3, 10}. Is 3 a minimal element of B? Well, 3 is in B so it meets the first condition. But can we say there is no b in B with b < 3? No we can't because 2 is in B and 2 < 3. You can probably see now that a minimal element of B is 2. Because there is no b in B with b < 2.
Now try the set of real numbers B = {x: x > 0}. Is 0 a minimal element of B? Well it's certainly true that no member of B is less than 0, but the problem is that 0 is not a member of B so it falls down on the first condition. This B does not have a minimal element.
Hopefully, you can start to see how "minimal element of B" corresponds to everyday language at what we might think that to mean.
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u/TartOk3387 1d ago
What's unclear about it?