Union of Dynkin systems is not a Dynkin system (counterexample)
If $X$ is a set and $mathcal{D_n}subset mathcal{P}(X)$ are Dynkin systems, then the union $displaystyle bigcup_{n} mathcal{D_n}$ is not always a Dynkin system. I need to find a counterexample for that, can anyone give me a hint?
measure-theory elementary-set-theory examples-counterexamples
edited 2 days ago
Davide Giraudo
125k16150260
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
add a comment |
If $X$ is a set and $mathcal{D_n}subset mathcal{P}(X)$ are Dynkin systems, then the union $displaystyle bigcup_{n} mathcal{D_n}$ is not always a Dynkin system. I need to find a counterexample for that, can anyone give me a hint?
measure-theory elementary-set-theory examples-counterexamples
edited 2 days ago
Davide Giraudo
125k16150260
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
add a comment |
If $X$ is a set and $mathcal{D_n}subset mathcal{P}(X)$ are Dynkin systems, then the union $displaystyle bigcup_{n} mathcal{D_n}$ is not always a Dynkin system. I need to find a counterexample for that, can anyone give me a hint?
measure-theory elementary-set-theory examples-counterexamples
edited 2 days ago
Davide Giraudo
125k16150260
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
If $X$ is a set and $mathcal{D_n}subset mathcal{P}(X)$ are Dynkin systems, then the union $displaystyle bigcup_{n} mathcal{D_n}$ is not always a Dynkin system. I need to find a counterexample for that, can anyone give me a hint?
measure-theory elementary-set-theory examples-counterexamples
measure-theory elementary-set-theory examples-counterexamples
edited 2 days ago
Davide Giraudo
125k16150260
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
edited 2 days ago
Davide Giraudo
125k16150260
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
edited 2 days ago
Davide Giraudo
125k16150260
edited 2 days ago
Davide Giraudo
125k16150260
edited 2 days ago
Davide Giraudo
125k16150260
125k16150260
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
837
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ is not a Dynkin system since ${1,2}cup{3,4}notin mathcal D_1cup mathcal D_2$, but these are disjoint sets.
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
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%2f2177691%2funion-of-dynkin-systems-is-not-a-dynkin-system-counterexample%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
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ is not a Dynkin system since ${1,2}cup{3,4}notin mathcal D_1cup mathcal D_2$, but these are disjoint sets.
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
add a comment |
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ is not a Dynkin system since ${1,2}cup{3,4}notin mathcal D_1cup mathcal D_2$, but these are disjoint sets.
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
add a comment |
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ is not a Dynkin system since ${1,2}cup{3,4}notin mathcal D_1cup mathcal D_2$, but these are disjoint sets.
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ is not a Dynkin system since ${1,2}cup{3,4}notin mathcal D_1cup mathcal D_2$, but these are disjoint sets.
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
edited Mar 8 '17 at 16:25
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
answered Mar 8 '17 at 16:17
NChNCh
6,2882723
6,2882723
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
add a comment |
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
– dimvolt
Mar 8 '17 at 16:23
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
Oh, cup vs cap are so close :) Thank you, i'll correct it.
– NCh
Mar 8 '17 at 16:25
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%2f2177691%2funion-of-dynkin-systems-is-not-a-dynkin-system-counterexample%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
