Cartesian Product as a tree

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












For any Cartesian product (at least for finite number of sets), I can associate it with a tree where the leaves are the sets and each node is the Cartesian product of node directly below it. For example, $(A times B times C) times (D times (E times F))$ is the root of the following tree



enter image description here



In fact, if $prod_i_1 prod_i_2 cdots prod_i_n X_i_1 i_2 ... i_n$ a finitely nested Cartesian product, then I can always associate such expression with a tree with finite height (if one accepts that a node can have infinite number of child nodes) and vice versa.

My question: is it possible to extend it to trees with infinite height? What would an infinitely nested Cartesian product look like?




elementary-set-theory trees




share|cite|improve this question




















  • Surely it would just look like an infinite tree?
    – Benedict Randall Shaw
    Aug 23 at 7:13










  • Sure, but I also want to know is there a set construction of such infinitely nested Cartesian product (if exist)?
    – Vinh Khang
    Aug 23 at 7:16










  • Also, it seems like I can never get to the root if I build up from the leaves, so it might not be even a tree?
    – Vinh Khang
    Aug 23 at 7:29










  • Note that this tree structure really isn't about Cartesian products at all, but simply abstract operations parenthesized in a particular way.
    – Greg Martin
    Aug 23 at 7:31










  • Indeed, I was trying to prove that any 2 "parenthesizations" of Cartesian products is in bijective correspondence, but then I got stuck at the case when the tree height goes to infinite because I do not know what would infinite tree would be like
    – Vinh Khang
    Aug 23 at 7:39















up vote
0
down vote

favorite












For any Cartesian product (at least for finite number of sets), I can associate it with a tree where the leaves are the sets and each node is the Cartesian product of node directly below it. For example, $(A times B times C) times (D times (E times F))$ is the root of the following tree



enter image description here



In fact, if $prod_i_1 prod_i_2 cdots prod_i_n X_i_1 i_2 ... i_n$ a finitely nested Cartesian product, then I can always associate such expression with a tree with finite height (if one accepts that a node can have infinite number of child nodes) and vice versa.

My question: is it possible to extend it to trees with infinite height? What would an infinitely nested Cartesian product look like?




elementary-set-theory trees




share|cite|improve this question




















  • Surely it would just look like an infinite tree?
    – Benedict Randall Shaw
    Aug 23 at 7:13










  • Sure, but I also want to know is there a set construction of such infinitely nested Cartesian product (if exist)?
    – Vinh Khang
    Aug 23 at 7:16










  • Also, it seems like I can never get to the root if I build up from the leaves, so it might not be even a tree?
    – Vinh Khang
    Aug 23 at 7:29










  • Note that this tree structure really isn't about Cartesian products at all, but simply abstract operations parenthesized in a particular way.
    – Greg Martin
    Aug 23 at 7:31










  • Indeed, I was trying to prove that any 2 "parenthesizations" of Cartesian products is in bijective correspondence, but then I got stuck at the case when the tree height goes to infinite because I do not know what would infinite tree would be like
    – Vinh Khang
    Aug 23 at 7:39













up vote
0
down vote

favorite









up vote
0
down vote

favorite











For any Cartesian product (at least for finite number of sets), I can associate it with a tree where the leaves are the sets and each node is the Cartesian product of node directly below it. For example, $(A times B times C) times (D times (E times F))$ is the root of the following tree



enter image description here



In fact, if $prod_i_1 prod_i_2 cdots prod_i_n X_i_1 i_2 ... i_n$ a finitely nested Cartesian product, then I can always associate such expression with a tree with finite height (if one accepts that a node can have infinite number of child nodes) and vice versa.

My question: is it possible to extend it to trees with infinite height? What would an infinitely nested Cartesian product look like?




elementary-set-theory trees




share|cite|improve this question












For any Cartesian product (at least for finite number of sets), I can associate it with a tree where the leaves are the sets and each node is the Cartesian product of node directly below it. For example, $(A times B times C) times (D times (E times F))$ is the root of the following tree



enter image description here



In fact, if $prod_i_1 prod_i_2 cdots prod_i_n X_i_1 i_2 ... i_n$ a finitely nested Cartesian product, then I can always associate such expression with a tree with finite height (if one accepts that a node can have infinite number of child nodes) and vice versa.

My question: is it possible to extend it to trees with infinite height? What would an infinitely nested Cartesian product look like?





elementary-set-theory trees





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 23 at 7:08









Vinh Khang

767




767











  • Surely it would just look like an infinite tree?
    – Benedict Randall Shaw
    Aug 23 at 7:13










  • Sure, but I also want to know is there a set construction of such infinitely nested Cartesian product (if exist)?
    – Vinh Khang
    Aug 23 at 7:16










  • Also, it seems like I can never get to the root if I build up from the leaves, so it might not be even a tree?
    – Vinh Khang
    Aug 23 at 7:29










  • Note that this tree structure really isn't about Cartesian products at all, but simply abstract operations parenthesized in a particular way.
    – Greg Martin
    Aug 23 at 7:31










  • Indeed, I was trying to prove that any 2 "parenthesizations" of Cartesian products is in bijective correspondence, but then I got stuck at the case when the tree height goes to infinite because I do not know what would infinite tree would be like
    – Vinh Khang
    Aug 23 at 7:39

















  • Surely it would just look like an infinite tree?
    – Benedict Randall Shaw
    Aug 23 at 7:13










  • Sure, but I also want to know is there a set construction of such infinitely nested Cartesian product (if exist)?
    – Vinh Khang
    Aug 23 at 7:16










  • Also, it seems like I can never get to the root if I build up from the leaves, so it might not be even a tree?
    – Vinh Khang
    Aug 23 at 7:29










  • Note that this tree structure really isn't about Cartesian products at all, but simply abstract operations parenthesized in a particular way.
    – Greg Martin
    Aug 23 at 7:31










  • Indeed, I was trying to prove that any 2 "parenthesizations" of Cartesian products is in bijective correspondence, but then I got stuck at the case when the tree height goes to infinite because I do not know what would infinite tree would be like
    – Vinh Khang
    Aug 23 at 7:39
















Surely it would just look like an infinite tree?
– Benedict Randall Shaw
Aug 23 at 7:13




Surely it would just look like an infinite tree?
– Benedict Randall Shaw
Aug 23 at 7:13












Sure, but I also want to know is there a set construction of such infinitely nested Cartesian product (if exist)?
– Vinh Khang
Aug 23 at 7:16




Sure, but I also want to know is there a set construction of such infinitely nested Cartesian product (if exist)?
– Vinh Khang
Aug 23 at 7:16












Also, it seems like I can never get to the root if I build up from the leaves, so it might not be even a tree?
– Vinh Khang
Aug 23 at 7:29




Also, it seems like I can never get to the root if I build up from the leaves, so it might not be even a tree?
– Vinh Khang
Aug 23 at 7:29












Note that this tree structure really isn't about Cartesian products at all, but simply abstract operations parenthesized in a particular way.
– Greg Martin
Aug 23 at 7:31




Note that this tree structure really isn't about Cartesian products at all, but simply abstract operations parenthesized in a particular way.
– Greg Martin
Aug 23 at 7:31












Indeed, I was trying to prove that any 2 "parenthesizations" of Cartesian products is in bijective correspondence, but then I got stuck at the case when the tree height goes to infinite because I do not know what would infinite tree would be like
– Vinh Khang
Aug 23 at 7:39





Indeed, I was trying to prove that any 2 "parenthesizations" of Cartesian products is in bijective correspondence, but then I got stuck at the case when the tree height goes to infinite because I do not know what would infinite tree would be like
– Vinh Khang
Aug 23 at 7:39
















active

oldest

votes











Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);













 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2891798%2fcartesian-product-as-a-tree%23new-answer', 'question_page');

);

Post as a guest






















active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2891798%2fcartesian-product-as-a-tree%23new-answer', 'question_page');

);

Post as a guest
































































-

這個網誌中的熱門文章

How to combine Bézier curves to a surface?

圖片
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
閱讀完整內容

Mutual Information Always Non-negative

圖片
Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 '12 at 15:38 Cameron Buie 83.6k 7 71 154 asked Jun 17 '12 at 15:30 Omri 359 3 13 3 Convexity of the function $tmapsto tlog t$. – Did Jun 17 '12 at 15:33 In addition, the convexity properties require the coefficients in the linear combination sum 1. Then, as p(x,y) is a probability distribution, it fullfits such condition. – Francisco Javier Delgado Ceped Aug 10 at 18:44 add a comment  |  up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 ...
閱讀完整內容

Why am i infinitely getting the same tweet with the Twitter Search API?

圖片
up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia...
閱讀完整內容