Mixing
Proof verification- $ sigma $ - Algebra, Algebra, Ring
$begingroup$
I want to prove following for a set $ X neq emptyset $ :
$ M subset P(X) $, where $ P(X) $ is the power set.
1) Any Ring is an Algebra
2) Any Algebra is a Ring
3) Any Ring is a $ sigma $ -Algebra
4) Any Algebra is a $ sigma $- Algebra
5) Any $ sigma $- Algebra is an Algebra
For a set $ X neq emptyset $, $M$ is
a) A Ring if
$ emptyset in M$
$ A,B in M rightarrow A cup B in M $
$ A,B in M rightarrow B backslash A in M $
b) An Algebra, if $ M$ Ring , and $ X in M $
c) A $ sigma $-Algebra if
$ X in M $
$ A in M rightarrow X backslash A in M $
$ (A_i)_{i in mathbb{N}} in M rightarrow cup_{i=0}^{ infty} A_i in M $
for 1)
For $ A subset X $ is M:={ emptyset , A} a Ring, but for $ A neq X$ not an Algebra.
2)
let be $ A,B in M $
$ A backslash B = ( A^c cup B )^c in M $
So M is also a ring.
3)
I guess I can use the same example as for 1)?
4) let be $ X= mathbb{N} , M:= { A subset X : A $ or $ A^c $ finite $}$
is this a right example for an Alegbra , which is not a s.Algeba?
5)
$A,B in M $
$ X= X backslash emptyset in M $
$ Abackslash B = A cap B^c = ( A^c cup B)^c in M $
any union you can discribe as :
$ A cup B = A cup B cup emptyset cup ...emptyset in M $
so any s. Algebra is also an Algebra.
are my arguments correct and formally right? any adjustments?
Appreciate any of your help !
real-analysis measure-theory proof-verification elementary-set-theory
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
asked Jan 23 at 13:51
wondering1123wondering1123
14911
$endgroup$
add a comment |
$begingroup$
I want to prove following for a set $ X neq emptyset $ :
$ M subset P(X) $, where $ P(X) $ is the power set.
1) Any Ring is an Algebra
2) Any Algebra is a Ring
3) Any Ring is a $ sigma $ -Algebra
4) Any Algebra is a $ sigma $- Algebra
5) Any $ sigma $- Algebra is an Algebra
For a set $ X neq emptyset $, $M$ is
a) A Ring if
$ emptyset in M$
$ A,B in M rightarrow A cup B in M $
$ A,B in M rightarrow B backslash A in M $
b) An Algebra, if $ M$ Ring , and $ X in M $
c) A $ sigma $-Algebra if
$ X in M $
$ A in M rightarrow X backslash A in M $
$ (A_i)_{i in mathbb{N}} in M rightarrow cup_{i=0}^{ infty} A_i in M $
for 1)
For $ A subset X $ is M:={ emptyset , A} a Ring, but for $ A neq X$ not an Algebra.
2)
let be $ A,B in M $
$ A backslash B = ( A^c cup B )^c in M $
So M is also a ring.
3)
I guess I can use the same example as for 1)?
4) let be $ X= mathbb{N} , M:= { A subset X : A $ or $ A^c $ finite $}$
is this a right example for an Alegbra , which is not a s.Algeba?
5)
$A,B in M $
$ X= X backslash emptyset in M $
$ Abackslash B = A cap B^c = ( A^c cup B)^c in M $
any union you can discribe as :
$ A cup B = A cup B cup emptyset cup ...emptyset in M $
so any s. Algebra is also an Algebra.
are my arguments correct and formally right? any adjustments?
Appreciate any of your help !
real-analysis measure-theory proof-verification elementary-set-theory
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
asked Jan 23 at 13:51
wondering1123wondering1123
14911
$endgroup$
$begingroup$
What is the role of the set $;X;$ in all those question ...??
$endgroup$
– DonAntonio
Jan 23 at 13:58
$begingroup$
it's a random set..if that's what you mean.?
$endgroup$
– wondering1123
Jan 23 at 16:47
$begingroup$
Do you see how to go from boolean algebra on $M $ to $sigma$-algebra on $M$ ? $A . B = A cap B$ and $A + B =...$
$endgroup$
– reuns
Jan 23 at 22:03
add a comment |
$begingroup$
I want to prove following for a set $ X neq emptyset $ :
$ M subset P(X) $, where $ P(X) $ is the power set.
1) Any Ring is an Algebra
2) Any Algebra is a Ring
3) Any Ring is a $ sigma $ -Algebra
4) Any Algebra is a $ sigma $- Algebra
5) Any $ sigma $- Algebra is an Algebra
For a set $ X neq emptyset $, $M$ is
a) A Ring if
$ emptyset in M$
$ A,B in M rightarrow A cup B in M $
$ A,B in M rightarrow B backslash A in M $
b) An Algebra, if $ M$ Ring , and $ X in M $
c) A $ sigma $-Algebra if
$ X in M $
$ A in M rightarrow X backslash A in M $
$ (A_i)_{i in mathbb{N}} in M rightarrow cup_{i=0}^{ infty} A_i in M $
for 1)
For $ A subset X $ is M:={ emptyset , A} a Ring, but for $ A neq X$ not an Algebra.
2)
let be $ A,B in M $
$ A backslash B = ( A^c cup B )^c in M $
So M is also a ring.
3)
I guess I can use the same example as for 1)?
4) let be $ X= mathbb{N} , M:= { A subset X : A $ or $ A^c $ finite $}$
is this a right example for an Alegbra , which is not a s.Algeba?
5)
$A,B in M $
$ X= X backslash emptyset in M $
$ Abackslash B = A cap B^c = ( A^c cup B)^c in M $
any union you can discribe as :
$ A cup B = A cup B cup emptyset cup ...emptyset in M $
so any s. Algebra is also an Algebra.
are my arguments correct and formally right? any adjustments?
Appreciate any of your help !
real-analysis measure-theory proof-verification elementary-set-theory
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
asked Jan 23 at 13:51
wondering1123wondering1123
14911
$endgroup$
I want to prove following for a set $ X neq emptyset $ :
$ M subset P(X) $, where $ P(X) $ is the power set.
1) Any Ring is an Algebra
2) Any Algebra is a Ring
3) Any Ring is a $ sigma $ -Algebra
4) Any Algebra is a $ sigma $- Algebra
5) Any $ sigma $- Algebra is an Algebra
For a set $ X neq emptyset $, $M$ is
a) A Ring if
$ emptyset in M$
$ A,B in M rightarrow A cup B in M $
$ A,B in M rightarrow B backslash A in M $
b) An Algebra, if $ M$ Ring , and $ X in M $
c) A $ sigma $-Algebra if
$ X in M $
$ A in M rightarrow X backslash A in M $
$ (A_i)_{i in mathbb{N}} in M rightarrow cup_{i=0}^{ infty} A_i in M $
for 1)
For $ A subset X $ is M:={ emptyset , A} a Ring, but for $ A neq X$ not an Algebra.
2)
let be $ A,B in M $
$ A backslash B = ( A^c cup B )^c in M $
So M is also a ring.
3)
I guess I can use the same example as for 1)?
4) let be $ X= mathbb{N} , M:= { A subset X : A $ or $ A^c $ finite $}$
is this a right example for an Alegbra , which is not a s.Algeba?
5)
$A,B in M $
$ X= X backslash emptyset in M $
$ Abackslash B = A cap B^c = ( A^c cup B)^c in M $
any union you can discribe as :
$ A cup B = A cup B cup emptyset cup ...emptyset in M $
so any s. Algebra is also an Algebra.
are my arguments correct and formally right? any adjustments?
Appreciate any of your help !
real-analysis measure-theory proof-verification elementary-set-theory
real-analysis measure-theory proof-verification elementary-set-theory
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
asked Jan 23 at 13:51
wondering1123wondering1123
14911
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
asked Jan 23 at 13:51
wondering1123wondering1123
14911
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
edited Jan 23 at 22:49
Andrés E. Caicedo
65.7k8160250
65.7k8160250
asked Jan 23 at 13:51
wondering1123wondering1123
14911
asked Jan 23 at 13:51
wondering1123wondering1123
14911
asked Jan 23 at 13:51
wondering1123wondering1123
14911
14911
$begingroup$
What is the role of the set $;X;$ in all those question ...??
$endgroup$
– DonAntonio
Jan 23 at 13:58
$begingroup$
it's a random set..if that's what you mean.?
$endgroup$
– wondering1123
Jan 23 at 16:47
$begingroup$
Do you see how to go from boolean algebra on $M $ to $sigma$-algebra on $M$ ? $A . B = A cap B$ and $A + B =...$
$endgroup$
– reuns
Jan 23 at 22:03
add a comment |
$begingroup$
What is the role of the set $;X;$ in all those question ...??
$endgroup$
– DonAntonio
Jan 23 at 13:58
$begingroup$
it's a random set..if that's what you mean.?
$endgroup$
– wondering1123
Jan 23 at 16:47
$begingroup$
Do you see how to go from boolean algebra on $M $ to $sigma$-algebra on $M$ ? $A . B = A cap B$ and $A + B =...$
$endgroup$
– reuns
Jan 23 at 22:03
$begingroup$
What is the role of the set $;X;$ in all those question ...??
$endgroup$
– DonAntonio
Jan 23 at 13:58
$begingroup$
What is the role of the set $;X;$ in all those question ...??
$endgroup$
– DonAntonio
Jan 23 at 13:58
$begingroup$
it's a random set..if that's what you mean.?
$endgroup$
– wondering1123
Jan 23 at 16:47
$begingroup$
it's a random set..if that's what you mean.?
$endgroup$
– wondering1123
Jan 23 at 16:47
$begingroup$
Do you see how to go from boolean algebra on $M $ to $sigma$-algebra on $M$ ? $A . B = A cap B$ and $A + B =...$
$endgroup$
– reuns
Jan 23 at 22:03
$begingroup$
Do you see how to go from boolean algebra on $M $ to $sigma$-algebra on $M$ ? $A . B = A cap B$ and $A + B =...$
$endgroup$
– reuns
Jan 23 at 22:03
add a comment |
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$begingroup$
What is the role of the set $;X;$ in all those question ...??
$endgroup$
– DonAntonio
Jan 23 at 13:58
$begingroup$
it's a random set..if that's what you mean.?
$endgroup$
– wondering1123
Jan 23 at 16:47
$begingroup$
Do you see how to go from boolean algebra on $M $ to $sigma$-algebra on $M$ ? $A . B = A cap B$ and $A + B =...$
$endgroup$
– reuns
Jan 23 at 22:03