Union (set theory) Puns

The union of two sets A and B is the set of elements which are in A, in B, or in both A and B.[2] In symbols,

A ∪ B = { x : x ∈ A  or  x ∈ B } 

For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is:

A = {x is an even integer larger than 1}
B = {x is an odd integer larger than 1}
A ∪ B = { 2 , 3 , 4 , 5 , 6 , … } 

As another example, the number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of even numbers {2, 4, 6, 8, 10, ...}, because 9 is neither prime nor even.

Sets cannot have duplicate elements,[3][4] so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents.

The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A.[8] In symbols:

x ∈ ⋃ M ⟺ ∃ A ∈ M ,   x ∈ A .

This idea subsumes the preceding sections—for example, A ∪ B ∪ C is the union of the collection {A, B, C}. Also, if M is the empty collection, then the union of M is the empty set. Notations

The notation for the general concept can vary considerably. For a finite union of sets S 1 , S 2 , S 3 , … , S n one often writes S 1 ∪ S 2 ∪ S 3 ∪ ⋯ ∪ S n S_n or ^n ⋃ ^i ^= ^1 Si Various common notations for arbitrary unions include ⋃ M, ⋃ ^A ^∈ ^M A , and ⋃ ^i ^∈ ^I A i . The last of these notations refers to the union of the collection { A i : i ∈ I } , where I is an index set and i is a set for every i ∈ I. In the case that the index set I is the set of natural numbers, one uses the notation ^∞ ⋃ ^i = 1 Ai , which is analogous to that of the infinite sums in series.[8]

When the symbol "∪" is placed before other symbols (instead of between them), it is usually rendered as a larger size

This question came to me while working on a hw assignment (but not an actual hw problem), basically: Let 𝛺 be a set and ∁ be a class of subsets of 𝛺. If we have two sets A, B , and we have that A∪B ∈ ∁, does that imply that A ∈ ∁ and B ∈ ∁?

I couldn't think of any counterexample, and I'm still thinking about whether this is true or not. Any help is appreciated.

So as I understand, something like this

data List a = Nil | Cons a (List a)

is considered a sum type because it is a sort of set theory "sum" or "disjoint union" of the two constructors. A disjoint union, as I understand, is just when you have a union of sets that have nothing shared; you "summed" the sets. So we imagine our List above a sort of disjoint union of Nil and Cons a (List a). Correct? What gets confusing is from a Wikipedia article on Algebraic data types :

>In set theory the equivalent of a sum type is a disjoint union, a set whose elements are pairs consisting of a tag (equivalent to a constructor) and an object of a type corresponding to the tag (equivalent to the constructor arguments).

The disjoint union article was fairly understandable, but I'm not making the connection between my List above and this quote above about tags and constructors and constructor arguments. Nice if someone could shine some light on this for me.

I need to prove that this is true for any series of groups (A₀,A₁,A₂,...).

I don't want to whole solution, I just want to understand how to perform this union of intersections and intersection of unions so I can even begin to answer this question.

Say my sets are A₀={0}, A₁={0,1}, A₂={0,1,2} and so on, just for illustration purposes.

If I want to find right side of this expression, the intersection of unions. The way I understand this, Is I find the intersection for I=0 of the union of all groups {A₁,A₂,...}, and then intersect that with the union of all groups {A₂,A₃,...}, and so on.

So with these illustration sets the right side will be just a set of all the positive integers? Am I on the right track even?

And then with the left side. I find the union for I=0 of the intersection of all groups {A₁,A₂,...}, which unless I'm completely off base is just the union of the singleton {1}, so the singleton {1}. And then I do the same for I=1 which will give me the singleton {2} and so on.

And then this side is also all the positive integers.

This seems wrong to me.

Problem: https://imgur.com/a/etSH7WK

So what I did was calculate A1 = [1,3] A2 = [1/2,3] A3 = [1/3,3] this will keep going on forever and if we do the union of all families my answer became ]0,3]

I did not include the 0 because 1/n cannot equal to 0 but the answer shows the union as [0,3] meaning that 0 is an element of the families. I believe the answer to be wrong but Im not sure if Im missing something

I was thinking about the surreal numbers, which is a collection too large to be a set (they make up a “proper class”, just as the collection of all sets is a proper class). Usually this is taken to imply that there is no way to extend them any further.

But then I started wondering if that’s really the case. Perhaps there is a way to develop a consistent theory of classes that are too large to be sets, and thus also a way to quantify how large they are (which would lead to new ordinals larger than any set-theoretic ordinal, and therefore an extension of the surreal numbers as well).

Thus we could speak of two non-set proper classes, one having strictly larger cardinality than the other. Intuitively this makes sense. Consider the class of all subclasses of the surreal numbers; there is no way to put them into one-to-one correspondence with the surreals themselves, and this strongly implies the former class is actually larger.

I’m not sure if it’s possible to build a rigorous and consistent theory of these objects, but it might be worth looking into.

We could, for instance, define a hyperset as a new type of object, which can be either plain or fancy. A plain hyperset contains only sets as members, but has too high cardinality to itself be a set—for example, the collection of all sets is a hyperset, as are many other existing examples of proper classes that are not sets. A fancy hyperset is also allowed to contain other hypersets as members in addition to sets, but cannot contain itself, similar to the rule for sets. Likewise, the class of all hypersets is not a hyperset. “Hyperordinals” (which describe the cardinalities of hypersets, and are larger than any ordinal) could be defined too, and would serve as the “birthdays” for a new extension of the surreal numbers.

There might actually be multiple consistent theories/universes of such objects with different axioms, just like how set theory works with or without the axiom of choice.

We might even be able to extend things further, using the notion of n-sets. The 1-sets are just sets, the 2-sets are what we have referred to as hypersets up to now, and the 3-sets are still larger (with the class of all 2-sets being one example). As a rule, the collection of all n-sets would be a n+1-set, and a n-set cannot include any m-sets as members if m>n. Each tier would thus be built on top of the one before.

Has anyone proposed or discussed extending set theory in such a way before? If not,

I'm reading Digital Image Processing by Gonzalez and Woods. After introducing the typical set theory they introduce operations for grayscale images.

Let A={(x, y, z)} where (x, y) are coordinates and z is intensity, say z an integer in [0, 255]. Then define the complement

A^c = {(x, y, K-z)| (x, y, z) in A}

where K would be 255 in my case (8 bit grayscale image). Then the union of two sets is defined by

A U B = {max_z(a, b)| a in A, b in B}

The author then leaves it at that, not defining intersection.

I'm wondering if there is a complete set theory (complete in the sense of defining all the operations, not in the Godel incompleteness theorem sense) that uses these definitions and where I might read about such things.

I thought I'd chime in on this as its the most interesting part of the show.

I reviewed Pennyworth ("SKIFFLE GOTHIC") pretty early on on my blog.

https://corben-dallas.blogspot.com/2019/09/pennyworth-is-crown-as-entertaining-b.html

Loved the atmosphere, the players, the script, the background. Eventually stopped watching because of the Bruno Heller panto vibe.

But I got drawn back again mainly by the actors and the intriguing background. I read one of the other theories on here, that the UK was never united and I like that but I think I've got a a better explanation solution to it and if I'm right the show creators might be presenting a desperately needed historical correction.

If you really like your alternative history you'll know that coming a close 2nd to "What if the Nazis won WW2?" as a subject, is "What if Britain stayed out of WW1?"

The latter subject now has actual historians such as Niall Ferguson saying Britain should have not helped France, show here recently

https://youtu.be/PPiFYkDdR_E

and let Imperial Germany, the 2nd Reich, dominate Europe from 1914 onwards and everything since would be hunky dory sunshine and rainbows.

The argument goes that the Kaisers Germany was about to become a prosperous Social Democratic state and would therefore just create the European Union for continental Europe, early in 1914. Even better, according to these historians, Britain would get to keep it's Empire!

I think the the setting of Pennyworth might trying to counter this, by showing the results of Britain staying out of WW1.

How powerful is the German Reich in the world of Pennyworth? Thomas Wayne says “In 20 years Gotham City will be the Zurich of the Eastern Seaboard!”

In Pennyworth Britain stayed out in 1914 and the moral costs are obvious. The Empire has pretty much collapsed anyway, as it would, and the price of fighting against that has rotted British society from the core outwards. The Royal family obviously doesn't have the authority it once had as it probably lives in the shadow of its now glittering German cousins. The British Army is all powerful as the RAF would never exist and the boat crazy Kaiser in charge of European super power would probably insist the Royal Navy be significantly downsized (or else).

At least we got the airships! - [https://en.wikipedia.org/wiki/Imperial_Airship_Scheme](https:/

So there's two pieces of evidence I've got for this:

1: The design and uniforms of both the characters are using the Union color scheme, and the background also looks to be in line with the Union designs we've seen in the past

2: A lot of people have been complaining about how we know pretty much nothing about the Union. What if this was intentional on Roosterteeth's part? Rather than an infodump by one of the main cast, we're treated to a far more effective exposition by seeing the Union from the point of view of the enemy.

Honestly, I'm really hoping this is the case. Either way, it looks like we'll be learning more about the Union and I really can't wait

https://imgur.com/a/VKA3t6f
I graphed Px and saw areas of overlap to get the intersection, and drew rectangles in the graph to physically show the union. I don't believe it was necessary to do this, but it helped me. For the intersection, I got that the sets intersect at {(0,n): 0<=n<=1}, but this relies on x=0, which appears to violate I=(0,1]. That being said, Px = [0,x] x [0, 1/x], so I'm a little confused if it's okay or not.
For the unions, my answer was that it would be a collection of all the points inside of Px. Perhaps this could be written like {(n,1/n): 0<=n<=1}, but I don't know. I feel pretty confident in that I'm on the right track, but I don't know if my answers are actually right. Any help that you guys could give would be greatly appreciated.
Thank you!

I'm wondering how to do operations with empty sets in set theory as pure sets?

Say I have a set A = { { }, { { } } }. From my understanding A has a cardinality of 2 and a rank of 2. Since the first subset has a rank of 0 (empty set) and the second has a rank of 1 (a set with one subset which is an empty set).

Also say I have a different set B = { { { { } } }, { { } } } which has cardinality of 2 and a rank of 3.

What would I get if I do a union of A and B? would it be { { }, { { } }, { { { } } } } ?

What would I get if I do an intersection of A and B? Would I get { { { } } }?

So say we have some infinite family of countable sets [; A_i ;], indexed by [; N ;] . Then we know for all [; A_i ;] there must exist a bijection [; f_i: N \to A_i ;], and so we can biject this to [; N \times N ;] pretty easily, with the bijection [; g(x,y) = f_x(y) ;], and thus we have that a countable union of countable sets is countable. I don't see where we use AOC in here, yet my book (Tao's analysis) says I do. Any ideas?

Let there exist a line split into segments by an infinite number of points so that segments are ordered by size from smallest in length to largest and that successive segments are double the length of previous ones.

The line, L, is an infinite set and the line segments are subsets of that set. The set of all such subsets, when unpacked by using the axiom of union, gives the set of natural numbers.

How would you express this information in terms of set theory?

Lets say we want to find the exterior of the set [0,1]x[0,1], in the topology with base B = { {x}x(y-r,y+r) : x,y are from R-real numbers , r>0 also real} over R^2 .My question is can we do it "algebraically" or simply using set operations like union, intersection, without, etc. ? It is actually easy to do by drawing it on R^2 plane. But can we do it without drawing, meaning using only set operations?

This is not a homework question,post was made simply out of curiosity. I found it did not really work using exterior([0,1]x[0,1])=interior((RxR)\([0,1]x[0,1]))

I first came across this puzzle in 2016. I thought it was a well-known problem arising from the axiom of choice, but after showing it to several mathematicians who had not seen it before I realize it may not be as famous as I thought (hence why I am posting it here).

There is a house with 100 rooms, and each room contains countably many boxes indexed with the natural numbers. Each box contains a random real number, which is the same over all the rooms (that is, box n contains the same real number in every room).

100 set theorists play a game. Each person will go into a unique room and open as many boxes as they like (perhaps countably many) as long as they leave at least one box in their room unopened. Then, each of them need to pick an unopened box in their room, and guess what real number is inside of it.

In order to win, 99 of them need to guess correctly.

The mathematicians can discuss a strategy beforehand, but after they go into their respective rooms, no more communication is allowed. What is a 100% winning strategy for this seemingly impossible task?

Solution here.

>!Let S be the set of sequences of real numbers, and let ~ be the binary relation on S where {x_j} ~ {y_j} if the two sequences agree for all but finitely many terms. Observe that ~ is an equivalence relation. Using the axiom of choice, the mathematicians agree on a representative sequence from each equivalence class of S under the ~ relation.!<

>!Once the representative sequences have been chosen, the mathematicians go into their respective rooms. For 1 ≤ n ≤ 100, let s_n be the sequence of reals given by the contents of boxes n, 100+n, 200+n,... and so on. Player 1 starts by opening every box except for 1, 101, 201,..., player 2 opens every box except for 2, 102, 202,... etcetera. In this way, every player sees 99 sequences of real numbers, in particular, player p sees every s_n for n ≠ p.!<

>!Now, each mathematician looks at each of the 99 sequences that they can see, identifies which equivalence class each of them belongs to, and recalls the chosen representative sequence for every one of them. Then for each sequence, the mathematicians write down the greatest index at which each observed sequence does not agree with its corresponding representative (which exists by the definition of ~). In this way, each player p has written down 99 integers x_n for n ≠ p, where x_n is the greatest index for which the sequence s_n disagr

Hullo all!

The theorem I'm trying to prove is the following: if [; \mathscr{P}(A) \cup \mathscr{P}(B) = \mathscr{P}(A \cup B) ;] then either [; A \subseteq B ;] or [; B \subseteq A ;].

Here is my attempt so far:

Suppose [; \mathscr{P}(A) \cup \mathscr{P}(B) = \mathscr{P}(A \cup B) ;]. If [; A \subseteq B ;] is true, then clearly [; A \subseteq B ;] or [; B \subseteq A ;] will be true. Thus consider the case where [; A \nsubseteq B ;]. Thus we can find an [; x ;] such that [; x \in A ;] and [; x \notin B ;]. Let [; y ;] be an arbitrary element of [; B ;]. Hence we both have [; x \in A \cup B ;] and [; y \in A \cup B ;]. Therefore we can find a set [; C ;] which contains both [; x ;] and [; y ;] such that [; C \in \mathscr{P}(A \cup B) ;]. Since [; \mathscr{P}(A) \cup \mathscr{P}(B) = \mathscr{P}(A \cup B) ;], [; C \in \mathscr{P}(A) \cup \mathscr{P}(B) ;] and thus [; C \in \mathscr{P}(A) ;] or [; C \in \mathscr{P}(B) ;]. We will consider these two cases separately:

Case 1: [; C \in \mathscr{P}(A) ;]. Since [; y \in C ;], then [; y \in A ;].

Case 2: [; C \in \mathscr{P}(B) ;]. Since [; x \in C ;], then [; x \in B ;]. However this contradicts the fact that [; x \notin B ;].

Since [; y ;] was arbitrary, then [; B \subseteq A ;], and thus if [; \mathscr{P}(A) \cup \mathscr{P}(B) = \mathscr{P}(A \cup B) ;] then either [; A \subseteq B ;] or [; B \subseteq A ;].

My issue: In all of the proofs by cases I've done, all of the cases turned out to be true. However in this particular instance as you see there is a contradiction in case 2. From what I know, since it's a disjunction the statement [; C \in \mathscr{P}(A) ;] or [; C \in \mathscr{P}(B) ;] remains true so it shouldn't matter (please correct me if I'm wrong here). Unless there is something wrong with my approach to this problem? Please help me.

I'm trying to prove that f-1(V) of a complex function is measurable if and only if f is measurable. I think I need the fact in the title for one direction of the proof. V is any open set in C.

Ever since I learned about the Continuum Hypothesis, this has been bugging me: if you want to find a set with a cardinality between the Integers and the Real Numbers, can't you just use a union of the Integers + one additional non-Integer element?

It seems to me that the union of two disjoint non-empty sets should have a cardinality greater than that of either set. Does this not apply to unions involving transfinite sets? If not, why?

Edit: I think I'm just going to go the Finitism route and pretend Cantor never happened. I appreciate the replies though.

Not very good at set theory so just wanted to make sure im not making a dumb mistake:

If A is a subset of X and B is a subset of Y, then I think the set

(X product Y) - (A product B) is equal to

((X - A) product Y) union (X product (Y-B))

Late night before exams, just want to make sure my tired brain isn't making some oversight.

I'm a high school student, and in a few days I'm having a quiz in set theory. Most of the quiz is going to be rigorously proving whatever statements (or providing a counterexample) that are given to us using logic rules, argument forms, and set rules.

An example of a problem is proving that A∩B=A∩C ∧ A∪B=A∪C → B=C

The symbols on the quiz are going to be subsets, set equality, intersection, and union, and this is all we need to know for the quiz.

What resources would you recommend for practicing how to do this? I need a lot of practice to be ready for the quiz. I would also be fine if you guys can give me some proofs to practice with.

For the first question, consider the example of math, which has the millennial prize problems as well as numerous famous conjectures. Physics, too, has its own set of mysteries such as the pursuit of a unified theory or baryon asymmetry. Is there such a holy grail of big problems that haven’t been solved yet in statistics?

Secondly,, is the theory of stats an evolving subject or is it mostly already developed and set in stone? For example, both math and physics are ever-evolving and newer and improved theories are formulated quite frequently in them. Is it the same if statistics? Meaning when statisticians in university departments do research, do they work on developing new models, ideas, and theories or do they mostly just apply the same pre-established principles to different fields and subjects to gain insight in that particular subject rather than to gain insight in the theory of stats itself (eg: using stats to study medical phenomena rather than studying pure stats to improve statistical theory)? Or is research in statistics mostly just developing new and better models based on preexisting concepts (most notably in AI and machine learning these days)?

I hope these two questions make sense. Just curious about what cutting-edge research in statistics is like and if it fits my interests.

Hello! Doing the exercises in Abbott's Analysis 2nd Ed.

Exercise 1.2.4

&nbsp;

Here is my attempt at a solution:

&nbsp;

A_1 = {x : x=1 or x is a multiplicative combination of any number of 2's}

A_2 = {x : x is a multiplicative combination of any number of 2's and at least one 3}

A_3 = {x : x is a multiplicative combination of any number of 2's, 3's and at least one 5}

.

.

.

A_n = {x : x is a multiplicative combination of any number of 2's, 3's, 5's, 7's, ... (p_n-1)'s, and at least one p_n}

where p_n indicates the nth prime number.

We know that there will be an infinite number of sets, as there are an infinite number of prime numbers. Furthermore, there will be an infinite number of elements in each set, as there is no bound on the number of multiplicative combinations that can be created.

&nbsp;

Please let me know if this would be an acceptable solution, or if there is anything I could do to make it better. Thanks! I would also be interested to hear if anyone else has a suggestion for other methods to generate such a collection of sets.

&nbsp;

EDIT: formatting

The ectoplasmic goo in Ghostbusters 2 behaves very close to the description of the warp being reflective of the psychic environment it's in. Hell the statue of liberty coming to life may have been humanities first warp construct. Not sure how this could have eventually led to the warp drive we see in Event Horizon but I feel it could have. I think that it's likely that the emperor was assuming the form of Bill Murray's character, Peter Venkman. I mean there is no clear reason why Venkman is really kept around at the university. He doesn't even believe in ghosts, he taints the experimental data if it means he can bone a college girl the movie actually makes more sense if emps is using his charismatic aura to stick around and observe physicists studying the warp for the first time. Murray spends most of his time trolling people and chasing Sigourney Weaver which I feel is very in keeping with the emperor's character. Lend me a hand and lets try to draw some other connections.

Look at how Venkmen (Bill Murray) acts in this scene and tell me it jsut makes so much more sense if he's emps: https://www.youtube.com/watch?v=9S4cldkdCjE&ab_channel=FilmdogsOnlineOr when Sigourney Weavers character comes in: https://www.youtube.com/watch?v=lnVxVykHmCE&ab_channel=Ghostbusters

The psycho-reactive warpslime: https://www.youtube.com/watch?v=9qTsB3qfNUk&ab_channel=Ghostbusters

Also think about ground hog day: https://www.youtube.com/watch?v=6VF5P7qLaEQ&ab_channel=GeoffKrall was this in fact, Tzeentchs attempt to stick the emperor in a time loop? Is this the first battle between chaos and the emperor?

Edit: I realized that most of the public basically just forgot about the ghostbusters activities, or it was brushed off as some kind of mass hysteria. So it's entirely possible that physicists a hundred years later came across some of the work Egon and Dan Akroyds character did (possibly pointed out to the Event Horizon engineering team by emps) and thought the mathematics gelled well with some newer 21st century physics and/or the materials the ghostbusters were using had some interesting physical properties and out popped the warp drive. The engine

See related math SE post: https://math.stackexchange.com/questions/4268754/is-fattening-countable-sets-a-good-way-to-compare-their-relative-density-withi

Basically, in summary, to get countable subsets of an uncountable set to play well with measure theory, it seems reasonable to "fatten" those sets by replacing every element with a ball of some small radius. This way you can compare their relative measure, which can be used to, for example, make conclusions about the probability of randomly selecting an element of some countable subset of a countable set (e.g. the probability of randomly selecting an integer when uniformly sampling from the rational numbers).

Any explanations of problems with this procedure, or links to work in this direction that has already been done, would be appreciated.

Hey guys,

I have this problem on my homework: https://imgur.com/a/tJvOf

I am a bit confused in what is asking and where to even start. In my attempt, I stated that:

part A is equal to 2 because 2 is the lowest possible number and must be the intersection.

part b is equal to all real numbers because the set is infinite. I feel this is wrong and I don't properly understand why.

Any help is greatly appreciated!

Hi all,

I have the following problem:

http://i.imgur.com/8tJxmDy.png

And I'll be honest, I have no idea where to even start with it. Unions and intersections are something we haven't been over in lectures, and it's not covered particularly well in the book, so I've been working on intuition up until this point. This one has me stumped though, so any help would be much appreciated.