i have a problem in an exercie about sets and sets of sets
0
$begingroup$
I have a set named $A$. This set is located in a set of sets named $T$. Knowing that $A$ is in $T$, I used this trick to prove the following:
Since $E$ is in $T$ then $Acup overline{A}$ is in $T$ (since $E=Acup overline{A}$)
and so we have $A$ in $T$ and $A cup overline{A}$ is in $T$
from that we conclude that $overline{A}$ is in $T$.
Is that correct?
If I made a mistake somewhere or violated some sort of law in the science of sets please clarify and thanks.
elementary-set-theory
edited Dec 7 '18 at 18:52
M.G
2,2281134
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
$endgroup$
add a comment |
0
$begingroup$
I have a set named $A$. This set is located in a set of sets named $T$. Knowing that $A$ is in $T$, I used this trick to prove the following:
Since $E$ is in $T$ then $Acup overline{A}$ is in $T$ (since $E=Acup overline{A}$)
and so we have $A$ in $T$ and $A cup overline{A}$ is in $T$
from that we conclude that $overline{A}$ is in $T$.
Is that correct?
If I made a mistake somewhere or violated some sort of law in the science of sets please clarify and thanks.
elementary-set-theory
edited Dec 7 '18 at 18:52
M.G
2,2281134
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
$endgroup$
1
$begingroup$
It is rather distracting to have to read "AUAbar" rather than $Acup overline{A}$. Please check out this page for a primer on how to type with MathJax and $LaTeX$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:48
1
$begingroup$
As for your question, you have not given nearly enough description of the set $T$. If you have certain known properties about $T$ then perhaps you might be able to conclude what you want, however with no known properties about $T$ you cannot make such a claim. Suppose your universal set is $E={1,2}$. Suppose that $A={1}$. Finally suppose that $T={emptyset,{1},{1,2}}$. Notice that $overline{A}={2}$ is not an element of $T$ despite $Acup overline{A}={1,2}$ is an element of $T$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:51
$begingroup$
If you happened to know that $T$ was closed under set differences such as is the case for sigma-algebras and you know that your universal set $E$ were an element of $T$, then you would from that be able to conclude that $T$ were closed under set complementation as well., however again without knowing that $T$ were closed under set differences you would have no way of knowing whether or not $T$ were closed under complementation and it is possible (as my example above shows) that it is not.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:53
$begingroup$
thanks alot this really helped and yes T={X in P(E) with f-1(f(X))=X}
$endgroup$
– Racem Besbes
Dec 7 '18 at 19:18
add a comment |
0
0
0
$begingroup$
I have a set named $A$. This set is located in a set of sets named $T$. Knowing that $A$ is in $T$, I used this trick to prove the following:
Since $E$ is in $T$ then $Acup overline{A}$ is in $T$ (since $E=Acup overline{A}$)
and so we have $A$ in $T$ and $A cup overline{A}$ is in $T$
from that we conclude that $overline{A}$ is in $T$.
Is that correct?
If I made a mistake somewhere or violated some sort of law in the science of sets please clarify and thanks.
elementary-set-theory
edited Dec 7 '18 at 18:52
M.G
2,2281134
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
$endgroup$
I have a set named $A$. This set is located in a set of sets named $T$. Knowing that $A$ is in $T$, I used this trick to prove the following:
Since $E$ is in $T$ then $Acup overline{A}$ is in $T$ (since $E=Acup overline{A}$)
and so we have $A$ in $T$ and $A cup overline{A}$ is in $T$
from that we conclude that $overline{A}$ is in $T$.
Is that correct?
If I made a mistake somewhere or violated some sort of law in the science of sets please clarify and thanks.
elementary-set-theory
elementary-set-theory
edited Dec 7 '18 at 18:52
M.G
2,2281134
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
edited Dec 7 '18 at 18:52
M.G
2,2281134
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
edited Dec 7 '18 at 18:52
M.G
2,2281134
edited Dec 7 '18 at 18:52
M.G
2,2281134
edited Dec 7 '18 at 18:52
M.G
2,2281134
2,2281134
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
asked Dec 7 '18 at 18:45
Racem BesbesRacem Besbes
11
11
1
$begingroup$
It is rather distracting to have to read "AUAbar" rather than $Acup overline{A}$. Please check out this page for a primer on how to type with MathJax and $LaTeX$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:48
1
$begingroup$
As for your question, you have not given nearly enough description of the set $T$. If you have certain known properties about $T$ then perhaps you might be able to conclude what you want, however with no known properties about $T$ you cannot make such a claim. Suppose your universal set is $E={1,2}$. Suppose that $A={1}$. Finally suppose that $T={emptyset,{1},{1,2}}$. Notice that $overline{A}={2}$ is not an element of $T$ despite $Acup overline{A}={1,2}$ is an element of $T$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:51
$begingroup$
If you happened to know that $T$ was closed under set differences such as is the case for sigma-algebras and you know that your universal set $E$ were an element of $T$, then you would from that be able to conclude that $T$ were closed under set complementation as well., however again without knowing that $T$ were closed under set differences you would have no way of knowing whether or not $T$ were closed under complementation and it is possible (as my example above shows) that it is not.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:53
$begingroup$
thanks alot this really helped and yes T={X in P(E) with f-1(f(X))=X}
$endgroup$
– Racem Besbes
Dec 7 '18 at 19:18
add a comment |
1
$begingroup$
It is rather distracting to have to read "AUAbar" rather than $Acup overline{A}$. Please check out this page for a primer on how to type with MathJax and $LaTeX$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:48
1
$begingroup$
As for your question, you have not given nearly enough description of the set $T$. If you have certain known properties about $T$ then perhaps you might be able to conclude what you want, however with no known properties about $T$ you cannot make such a claim. Suppose your universal set is $E={1,2}$. Suppose that $A={1}$. Finally suppose that $T={emptyset,{1},{1,2}}$. Notice that $overline{A}={2}$ is not an element of $T$ despite $Acup overline{A}={1,2}$ is an element of $T$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:51
$begingroup$
If you happened to know that $T$ was closed under set differences such as is the case for sigma-algebras and you know that your universal set $E$ were an element of $T$, then you would from that be able to conclude that $T$ were closed under set complementation as well., however again without knowing that $T$ were closed under set differences you would have no way of knowing whether or not $T$ were closed under complementation and it is possible (as my example above shows) that it is not.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:53
$begingroup$
thanks alot this really helped and yes T={X in P(E) with f-1(f(X))=X}
$endgroup$
– Racem Besbes
Dec 7 '18 at 19:18
1
1
$begingroup$
It is rather distracting to have to read "AUAbar" rather than $Acup overline{A}$. Please check out this page for a primer on how to type with MathJax and $LaTeX$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:48
$begingroup$
It is rather distracting to have to read "AUAbar" rather than $Acup overline{A}$. Please check out this page for a primer on how to type with MathJax and $LaTeX$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:48
1
1
$begingroup$
As for your question, you have not given nearly enough description of the set $T$. If you have certain known properties about $T$ then perhaps you might be able to conclude what you want, however with no known properties about $T$ you cannot make such a claim. Suppose your universal set is $E={1,2}$. Suppose that $A={1}$. Finally suppose that $T={emptyset,{1},{1,2}}$. Notice that $overline{A}={2}$ is not an element of $T$ despite $Acup overline{A}={1,2}$ is an element of $T$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:51
$begingroup$
As for your question, you have not given nearly enough description of the set $T$. If you have certain known properties about $T$ then perhaps you might be able to conclude what you want, however with no known properties about $T$ you cannot make such a claim. Suppose your universal set is $E={1,2}$. Suppose that $A={1}$. Finally suppose that $T={emptyset,{1},{1,2}}$. Notice that $overline{A}={2}$ is not an element of $T$ despite $Acup overline{A}={1,2}$ is an element of $T$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:51
$begingroup$
If you happened to know that $T$ was closed under set differences such as is the case for sigma-algebras and you know that your universal set $E$ were an element of $T$, then you would from that be able to conclude that $T$ were closed under set complementation as well., however again without knowing that $T$ were closed under set differences you would have no way of knowing whether or not $T$ were closed under complementation and it is possible (as my example above shows) that it is not.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:53
$begingroup$
If you happened to know that $T$ was closed under set differences such as is the case for sigma-algebras and you know that your universal set $E$ were an element of $T$, then you would from that be able to conclude that $T$ were closed under set complementation as well., however again without knowing that $T$ were closed under set differences you would have no way of knowing whether or not $T$ were closed under complementation and it is possible (as my example above shows) that it is not.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:53
$begingroup$
thanks alot this really helped and yes T={X in P(E) with f-1(f(X))=X}
$endgroup$
– Racem Besbes
Dec 7 '18 at 19:18
$begingroup$
thanks alot this really helped and yes T={X in P(E) with f-1(f(X))=X}
$endgroup$
– Racem Besbes
Dec 7 '18 at 19:18
add a comment |
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1
$begingroup$
It is rather distracting to have to read "AUAbar" rather than $Acup overline{A}$. Please check out this page for a primer on how to type with MathJax and $LaTeX$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:48
1
$begingroup$
As for your question, you have not given nearly enough description of the set $T$. If you have certain known properties about $T$ then perhaps you might be able to conclude what you want, however with no known properties about $T$ you cannot make such a claim. Suppose your universal set is $E={1,2}$. Suppose that $A={1}$. Finally suppose that $T={emptyset,{1},{1,2}}$. Notice that $overline{A}={2}$ is not an element of $T$ despite $Acup overline{A}={1,2}$ is an element of $T$.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:51
$begingroup$
If you happened to know that $T$ was closed under set differences such as is the case for sigma-algebras and you know that your universal set $E$ were an element of $T$, then you would from that be able to conclude that $T$ were closed under set complementation as well., however again without knowing that $T$ were closed under set differences you would have no way of knowing whether or not $T$ were closed under complementation and it is possible (as my example above shows) that it is not.
$endgroup$
– JMoravitz
Dec 7 '18 at 18:53
$begingroup$
thanks alot this really helped and yes T={X in P(E) with f-1(f(X))=X}
$endgroup$
– Racem Besbes
Dec 7 '18 at 19:18