How to draw the cartesian product of two infinite sets?
If I have sets A and B with the cardinality of the set of Integers, making these sets denuermable sets by definition, how could they be drawn?
I have a rough idea as it would look like ordered pairs, just like a coordnate for a 2D grid, inside a little box surrounded by multiple boxes going in all directions holding different ordered pairs.
discrete-mathematics
asked Nov 30 at 0:59
Zdravstvuyte94
355
add a comment |
If I have sets A and B with the cardinality of the set of Integers, making these sets denuermable sets by definition, how could they be drawn?
I have a rough idea as it would look like ordered pairs, just like a coordnate for a 2D grid, inside a little box surrounded by multiple boxes going in all directions holding different ordered pairs.
discrete-mathematics
asked Nov 30 at 0:59
Zdravstvuyte94
355
1
It sounds like you are talking about a square lattice... but I can't be sure what you're getting at.
– JMoravitz
Nov 30 at 1:01
Why wouldn't the be drawn the same way $mathbb Z times mathbb Z$ would? You have an $x$ axis with all elements of $A$ listed in some order and a $y$ axiss with all the elements of $B$ listed in some order and the each element $(a,b)$ in the where the line paralell to the $y$ axis at $a$ on the $x$ axis intersets the line paralel to $x$ axis and $b$ on the $y$ axis.
– fleablood
Nov 30 at 7:03
add a comment |
If I have sets A and B with the cardinality of the set of Integers, making these sets denuermable sets by definition, how could they be drawn?
I have a rough idea as it would look like ordered pairs, just like a coordnate for a 2D grid, inside a little box surrounded by multiple boxes going in all directions holding different ordered pairs.
discrete-mathematics
asked Nov 30 at 0:59
Zdravstvuyte94
355
If I have sets A and B with the cardinality of the set of Integers, making these sets denuermable sets by definition, how could they be drawn?
I have a rough idea as it would look like ordered pairs, just like a coordnate for a 2D grid, inside a little box surrounded by multiple boxes going in all directions holding different ordered pairs.
discrete-mathematics
discrete-mathematics
asked Nov 30 at 0:59
Zdravstvuyte94
355
asked Nov 30 at 0:59
Zdravstvuyte94
355
asked Nov 30 at 0:59
Zdravstvuyte94
355
asked Nov 30 at 0:59
Zdravstvuyte94
355
asked Nov 30 at 0:59
Zdravstvuyte94
355
355
1
It sounds like you are talking about a square lattice... but I can't be sure what you're getting at.
– JMoravitz
Nov 30 at 1:01
Why wouldn't the be drawn the same way $mathbb Z times mathbb Z$ would? You have an $x$ axis with all elements of $A$ listed in some order and a $y$ axiss with all the elements of $B$ listed in some order and the each element $(a,b)$ in the where the line paralell to the $y$ axis at $a$ on the $x$ axis intersets the line paralel to $x$ axis and $b$ on the $y$ axis.
– fleablood
Nov 30 at 7:03
add a comment |
1
It sounds like you are talking about a square lattice... but I can't be sure what you're getting at.
– JMoravitz
Nov 30 at 1:01
Why wouldn't the be drawn the same way $mathbb Z times mathbb Z$ would? You have an $x$ axis with all elements of $A$ listed in some order and a $y$ axiss with all the elements of $B$ listed in some order and the each element $(a,b)$ in the where the line paralell to the $y$ axis at $a$ on the $x$ axis intersets the line paralel to $x$ axis and $b$ on the $y$ axis.
– fleablood
Nov 30 at 7:03
1
1
It sounds like you are talking about a square lattice... but I can't be sure what you're getting at.
– JMoravitz
Nov 30 at 1:01
It sounds like you are talking about a square lattice... but I can't be sure what you're getting at.
– JMoravitz
Nov 30 at 1:01
Why wouldn't the be drawn the same way $mathbb Z times mathbb Z$ would? You have an $x$ axis with all elements of $A$ listed in some order and a $y$ axiss with all the elements of $B$ listed in some order and the each element $(a,b)$ in the where the line paralell to the $y$ axis at $a$ on the $x$ axis intersets the line paralel to $x$ axis and $b$ on the $y$ axis.
– fleablood
Nov 30 at 7:03
Why wouldn't the be drawn the same way $mathbb Z times mathbb Z$ would? You have an $x$ axis with all elements of $A$ listed in some order and a $y$ axiss with all the elements of $B$ listed in some order and the each element $(a,b)$ in the where the line paralell to the $y$ axis at $a$ on the $x$ axis intersets the line paralel to $x$ axis and $b$ on the $y$ axis.
– fleablood
Nov 30 at 7:03
add a comment |
1 Answer
1
active
oldest
votes
List A as $a_1, a_2, a_3, ...$ and B as $b_1, b_2, b_3, ....$
Place for all positive integers k, $a_k$ on (k,0) and $b_k$ on (0,k).
You will see A laided out as positive integers on the x-axis
and B laided out as positive integers on the x-axis.
Consequently you'll see AxB laided out as all the
points (n,m) on the plane with positive integers n,m.
answered Nov 30 at 3:44
William Elliot
7,0702519
add a comment |
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',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019476%2fhow-to-draw-the-cartesian-product-of-two-infinite-sets%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
List A as $a_1, a_2, a_3, ...$ and B as $b_1, b_2, b_3, ....$
Place for all positive integers k, $a_k$ on (k,0) and $b_k$ on (0,k).
You will see A laided out as positive integers on the x-axis
and B laided out as positive integers on the x-axis.
Consequently you'll see AxB laided out as all the
points (n,m) on the plane with positive integers n,m.
answered Nov 30 at 3:44
William Elliot
7,0702519
add a comment |
List A as $a_1, a_2, a_3, ...$ and B as $b_1, b_2, b_3, ....$
Place for all positive integers k, $a_k$ on (k,0) and $b_k$ on (0,k).
You will see A laided out as positive integers on the x-axis
and B laided out as positive integers on the x-axis.
Consequently you'll see AxB laided out as all the
points (n,m) on the plane with positive integers n,m.
answered Nov 30 at 3:44
William Elliot
7,0702519
add a comment |
List A as $a_1, a_2, a_3, ...$ and B as $b_1, b_2, b_3, ....$
Place for all positive integers k, $a_k$ on (k,0) and $b_k$ on (0,k).
You will see A laided out as positive integers on the x-axis
and B laided out as positive integers on the x-axis.
Consequently you'll see AxB laided out as all the
points (n,m) on the plane with positive integers n,m.
answered Nov 30 at 3:44
William Elliot
7,0702519
List A as $a_1, a_2, a_3, ...$ and B as $b_1, b_2, b_3, ....$
Place for all positive integers k, $a_k$ on (k,0) and $b_k$ on (0,k).
You will see A laided out as positive integers on the x-axis
and B laided out as positive integers on the x-axis.
Consequently you'll see AxB laided out as all the
points (n,m) on the plane with positive integers n,m.
answered Nov 30 at 3:44
William Elliot
7,0702519
answered Nov 30 at 3:44
William Elliot
7,0702519
answered Nov 30 at 3:44
William Elliot
7,0702519
answered Nov 30 at 3:44
William Elliot
7,0702519
7,0702519
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019476%2fhow-to-draw-the-cartesian-product-of-two-infinite-sets%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
It sounds like you are talking about a square lattice... but I can't be sure what you're getting at.
– JMoravitz
Nov 30 at 1:01
Why wouldn't the be drawn the same way $mathbb Z times mathbb Z$ would? You have an $x$ axis with all elements of $A$ listed in some order and a $y$ axiss with all the elements of $B$ listed in some order and the each element $(a,b)$ in the where the line paralell to the $y$ axis at $a$ on the $x$ axis intersets the line paralel to $x$ axis and $b$ on the $y$ axis.
– fleablood
Nov 30 at 7:03