Union of Dynkin systems is not a Dynkin system (counterexample)
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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 Jan 5 at 16:48
Davide Giraudo
128k17156268
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
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add a comment |
$begingroup$
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 Jan 5 at 16:48
Davide Giraudo
128k17156268
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
$endgroup$
add a comment |
$begingroup$
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 Jan 5 at 16:48
Davide Giraudo
128k17156268
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
$endgroup$
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 Jan 5 at 16:48
Davide Giraudo
128k17156268
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
edited Jan 5 at 16:48
Davide Giraudo
128k17156268
asked Mar 8 '17 at 15:15
dimvoltdimvolt
837
edited Jan 5 at 16:48
Davide Giraudo
128k17156268
edited Jan 5 at 16:48
Davide Giraudo
128k17156268
edited Jan 5 at 16:48
Davide Giraudo
128k17156268
128k17156268
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
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$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ 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
7,0363825
$endgroup$
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ 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
7,0363825
$endgroup$
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
add a comment |
$begingroup$
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ 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
7,0363825
$endgroup$
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
add a comment |
$begingroup$
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ 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
7,0363825
$endgroup$
$X={1,2,3,4,5,6}$,
$$mathcal D_1=bigl{varnothing, X, {1,2}, {3,4,5,6}, {1,3}, {2,4,5,6}bigr}$$
$$mathcal D_2=bigl{varnothing, X, {3,4}, {1,2,5,6}, {3,5}, {1,2,4,6}bigr}$$
$mathcal D_1cup mathcal D_2$ 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
7,0363825
edited Mar 8 '17 at 16:25
answered Mar 8 '17 at 16:17
NChNCh
7,0363825
answered Mar 8 '17 at 16:17
NChNCh
7,0363825
answered Mar 8 '17 at 16:17
NChNCh
7,0363825
7,0363825
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
add a comment |
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
You probably mean ${1,2}cup{3,4}$ . Thank you for your answer!
$endgroup$
– dimvolt
Mar 8 '17 at 16:23
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
$begingroup$
Oh, cup vs cap are so close :) Thank you, i'll correct it.
$endgroup$
– NCh
Mar 8 '17 at 16:25
add a comment |
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