elementary set theory – Notation for set containing combinations of sets given by multinomial coefficient number of options.

Assume there are sets $A_1,A_2,dots,A_n$. Let $mleq n$ and now partition ${1,dots,n}$ into $m$ subsets $N_1,dots,N_m$. So, there are basically
$$sum_{l_1+dots+l_m = n} {{n}choose{l_1,dots,l_m}}$$
different ways to do this.
$$B_i = bigcap_{iin N_i} A_i cap bigcap_{iin{1,dots,n}setminus N_i} A_i^c.$$
I am interested in the set containing ${B_1,dots,B_m}$ for all possible partitions of ${1,dots,n}$ into $m$ sets. So something like
$$C={{B_1,dots,B_m} | N_1,dots,N_m text{ partition of }{1,dots,n} text{ and } B_i text{ chosen as above}}.$$
Is there are more elegant way to define this $C$?

nt.number theory – Cyclotomic Numbers, Difference Sets

I have reading papers by Cunsheng Ding on Binary Cyclotomic Generators, Linear Complexity of Generalised Binary Sequences of Order 2. Since the topic is new to me understanding the text is quite difficult for me, can someone suggest some references and text, I had looked up the references given in the paper but still no clue how to go through it sequentially. I have to submit my report for the project by summarizing these papers. any lead related to it will be very helpful.

real analysis – Complete and Incomplete Sets

For $A, B$ $⊆$ $mathbb R^{n}$, let $A + B :=$ {$a + b| a ∈ A,b ∈ B$} and:

$A := mathbb R times {0} ⊂ mathbb R^{2} $ and $B :=$ {$(x,y) ∈ mathbb R^{2}| xy = 1$}

(a) Show that $A$ and $B$ are complete.

(b) Show that $A + B$ are not complete.

How do we show that each set is complete and their sum is not complete? I know the question is basic, but due to COVID, online lectures have been tough for me. Thank you very much.

algorithms – show the problem of find two subsets such that the difference of them of two sets is smaller than a value, is NP-Hard

As input, given two finite sets of integers X = {x1,…,xm},Y = {y1,…,yn} ⊆ Z, and a non-negative integer v ≥ 0. The goal is to decide if there are non-empty subsets S ⊆ [m] and T ⊆ [n] such thatenter image description here

How to show this problem is NP-Complete? I’m quite confused.
What I got so far is to reduct from subset sum problems, since the form is set to less than v. So I need to have 2v+1 subset sum problems to verify

set theory – A Set with Three Elements Equals 8 total sets?

So I was watching a lecture by Chomsky, and he mentioned “New Math” and how his 10 y/o daighter was learning Set Theory and Boolean Algebra. Now, I thought I had a handle on naive set theory, and was going to try to get involved in a little side project where I employ Predicate Calculus and HOL to sort of philosophically comment on internal qualia and how they are non-transferable between aware observers (Hard problem of consciousness). I am definitely not including my most likely maladaptive, and remedial syntax. I am okay with Predicate Calculus inasmuch as I (think) I can make well formed formulas, and even if not I still underatand the genetal syntax and I find it pretty fun to play around with, and I want to work my way up to HOL and Set Theory, so I can somehow solipsistically describe what we term “Reality” as an aggregation of internal qualia. What qualia are is problematic to define, because redness could be a quality, as could texture, but through gradations of color in a red stone, I get the qualia “rough stone texture.” so you can see where we could argue different sets of electromagnetic frequencies aggregated by perception are basically then subests of other qualia. Texture could ibclude kinesthetic qualia and visual qualia. And when we start with qualia, you get something along the lines of thrownness in existentialism, since if we start with qualia (which we all do, or if you want to tell me I currently am not utilizing qualia generated by my CNS I implore you to elaborate on exactly what mechanism does act as a constituent to my personal model of reality.) It implied an aware observer… And later qualia enabled us to discover that qualities of visual perception are electromagnetic in constitution, i.e. light.p

I have come a long way since my last (arguably dimwitted) post. I finished a complete introduction to logic because I LOVED it. Absorbed like a sponge.

Now back to the lecture. Chomsky said his daughter seemed to understand Set Theory pretty well, and he said that his Israeli friend was trying to teach basic set theory to Junior High kids and having an issue doing so. Now, here is what got me. He said he asked his daughter if “you have three items, how many sets do you have?” And she replied (he seemed to think correctly) 8 sets. She listed them. She had my same logic for assuming the null set was in every set (she saw the spaces in between members and concluded it had to be there, as did I. When I saw this was actually correct, I was ecstatic, even though that was a couple years ago now, and I find out another kid was on the same page), because when I was little I used to argue that 0 was God because it is all around us, and that I could always collect a whole ton of nothing. (I made a nothing club in third grade based on a primitive understanding of naive set theory cuz mom is data analyst and had fancy math books and I got in some trouble and got my game consoles taken away so that was my reading material! I started using 0/0 = x where x is any nimber to “prove” that every number equals every other number, and my lower grade math teachers got mad, but High School math teacher was totally in on the joke and said “Hey, by that logic, he is right” he said, with a big grin on his face, much to the chagrin of a female classmate who raised her hand to get him to tell me I was wrong.)

Now, I understand you can use material implication to equate x ∈ A Ξ xP(x) , and you can do the same via A = {x | xP(x)} where P is a predicate. I like using parentheses although I noticed most books (*** that I have read, a lot of sources do use parentheses though, and sometimes I only use them when i have a two place predicate and I want to connect xHx(Se), intending you to read that as “x such that x is H of/by/from/to (e is S)”) dealing just in logic avoid this, but I don’t see it as inherently wrong so long as it is clarified that it is a predicate and not a function… although i feel as if you could view “Concept Script” as a function that merely returns True or False but hell if I know. The meat and potatoes is the notation of Power Sets. That is the only way I could feasibly derive eight sets from three objects (couldn’t tell if he meant 3 objects, a set of three objects, or three singletons, so I assume he meant !a set of three things based on his enumeration), but iI feel like that is just because of a very dubious misappropriation of the notation 2ⁿ where the lower case n is a capital A as a superscript representing a set, and serving as an alternative to ℘(A). If I count the null set, the singletons, etc, I can play abstract subset gymnastica to try to get eight sets, but I see nothing confirming that this is viable as an approach except this:

The set of all subsets of a set A is called the power set of A and denoted as ℘(A) or
sometimes as 2^A
For example, if A = {a,b}, ℘(A) = {∅, {a}, {b}, {a,b}}

Is he referring to Power Sets? I feel like i am taking the 2^A thing too literally. If I do, a set of three items becomes 2^3, thus eight. But I dont think they want me to take the cardinality of the set and use that as the exponential of 2, like some bizarre set theory logarithm 😝

Fact check me on my use of cardinality as well.

Any guidance humbly welcome, but I just want to know the math and how it works. Please avoid my bizarre meanderings about childhood, and the hard problem of conscious as these ideas are pre-alpha phase. I need to understand set theory and HOL FIRST then we can move up to writing my take on the hard problem of consciousness. I want to see if I can relate it back to Munchausen/Agrippa’s Trilemma as I feel like they are connected.

All tags were intentional, as I am lookout for insight into all of these fields. Please, do not edit them assuming I am talking out of my ass. I believe type theory is where subclasses come in? Haven’t gotten that far…okay, I did, but then I forgot xD

I thought it had to do with Russel’s Paradox. I heard Frege had a mental breakdown at this, whereas Russell was more confident. Then subclasses and type theory emerged to combat the issue

But again main question… how did she get 8 sets? Power sets? My interpretation is too literal, no? Like how you shouldn’t take dy/dx as literal division, even though Grant Sanderson (3blue1brown) seems to think it is helpful when learning calculus.

dnd 5e – Should I buy one of the starter sets, or the core books?

dnd 5e – Should I buy one of the starter sets, or the core books? – Role-playing Games Stack Exchange

Limits of indicators of sets.

Suppose $A_n$ is a sequence of sets such that $lim_{ntoinfty}A_n = A$ where we make no assumption on the $A_n$ being increasing or decreasing. Is $lim_{nto infty}chi_{A_n} = chi_{A}$?

For example consider a sequence in the reals $x_n$ that has limit $x$ then define $A_n = [0,x_n]$. Is it true that $lim_{ntoinfty} chi_{[0,x_n]} = chi_{[0,x]}$?

If the example given does not hold what are some conditions such that it does?

Thanks :]

When I say $chi_S$ I mean the indicator of set $S$, maybe you have seen it as $mathbb{1}_S$ :]

Sets and orthogonality

Let ${v_1,v_2,dots,v_n}$ be a basis of vector space $V_n$
$V_1=span{v_1+v_2+dots+v_n}$ and $V_2={v=sum_{j=1}^nx_jv_j | sum_{j=1}^nx_j=0,x_jin R}$
Prove that $V=V_1bigoplus V_2$, which I suppose is to prove $V_1perp V_2$

Put this in $3D$ scenario and let the basis be standard, then $V_2$ is the plane $x+y+z=0$ that perpendicular to $V_1$ which is spanned by the vector $(1,1,1)$
therefore their union fills $R^3$
However, how do I explain this in general?

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