Mixing
Why a cylinder set with base in $mathbbR^n$ can have a base in $mathbbR^n+1$ as well?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let define a cylinder set as:
$$mathcalC(B_1 times dots times B_n)=x in mathbbR^infty$$
where $B_k in mathcalB(mathbbR)$ for $k=1,2,...,n$ with base in $mathbbR^n$.
But, we can write also:
$$mathcalC(B_1 times dots times B_n)=mathcalC(B_1 times
dots times B_ntimes mathbbR)$$
with base in $mathbbR^n+1$.
Why is this statement true?
measure-theory intuition
asked Jul 30 at 9:53
Melina
87416
add a comment |Â
up vote
0
down vote
favorite
Let define a cylinder set as:
$$mathcalC(B_1 times dots times B_n)=x in mathbbR^infty$$
where $B_k in mathcalB(mathbbR)$ for $k=1,2,...,n$ with base in $mathbbR^n$.
But, we can write also:
$$mathcalC(B_1 times dots times B_n)=mathcalC(B_1 times
dots times B_ntimes mathbbR)$$
with base in $mathbbR^n+1$.
Why is this statement true?
measure-theory intuition
asked Jul 30 at 9:53
Melina
87416
Have you made an attempt?
– Kavi Rama Murthy
Jul 30 at 9:54
@KaviRamaMurthy Yes I made an attempt to understand why is this true and I read also in the book of Shiryaev but I couldn't figure out.
– Melina
Jul 30 at 9:55
No proof is required. The equality is true by definition.
– Kavi Rama Murthy
Jul 30 at 10:02
1
Can you find an element in the set denoted on LHS that is not in the set denoted on RHS? Can you find an element in the set denoted on RHS that is not in the set denoted on LHS?
– drhab
Jul 30 at 10:07
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let define a cylinder set as:
$$mathcalC(B_1 times dots times B_n)=x in mathbbR^infty$$
where $B_k in mathcalB(mathbbR)$ for $k=1,2,...,n$ with base in $mathbbR^n$.
But, we can write also:
$$mathcalC(B_1 times dots times B_n)=mathcalC(B_1 times
dots times B_ntimes mathbbR)$$
with base in $mathbbR^n+1$.
Why is this statement true?
measure-theory intuition
asked Jul 30 at 9:53
Melina
87416
Let define a cylinder set as:
$$mathcalC(B_1 times dots times B_n)=x in mathbbR^infty$$
where $B_k in mathcalB(mathbbR)$ for $k=1,2,...,n$ with base in $mathbbR^n$.
But, we can write also:
$$mathcalC(B_1 times dots times B_n)=mathcalC(B_1 times
dots times B_ntimes mathbbR)$$
with base in $mathbbR^n+1$.
Why is this statement true?
measure-theory intuition
asked Jul 30 at 9:53
Melina
87416
asked Jul 30 at 9:53
Melina
87416
asked Jul 30 at 9:53
Melina
87416
asked Jul 30 at 9:53
Melina
87416
87416
Have you made an attempt?
– Kavi Rama Murthy
Jul 30 at 9:54
@KaviRamaMurthy Yes I made an attempt to understand why is this true and I read also in the book of Shiryaev but I couldn't figure out.
– Melina
Jul 30 at 9:55
No proof is required. The equality is true by definition.
– Kavi Rama Murthy
Jul 30 at 10:02
1
Can you find an element in the set denoted on LHS that is not in the set denoted on RHS? Can you find an element in the set denoted on RHS that is not in the set denoted on LHS?
– drhab
Jul 30 at 10:07
add a comment |Â
Have you made an attempt?
– Kavi Rama Murthy
Jul 30 at 9:54
@KaviRamaMurthy Yes I made an attempt to understand why is this true and I read also in the book of Shiryaev but I couldn't figure out.
– Melina
Jul 30 at 9:55
No proof is required. The equality is true by definition.
– Kavi Rama Murthy
Jul 30 at 10:02
1
Can you find an element in the set denoted on LHS that is not in the set denoted on RHS? Can you find an element in the set denoted on RHS that is not in the set denoted on LHS?
– drhab
Jul 30 at 10:07
Have you made an attempt?
– Kavi Rama Murthy
Jul 30 at 9:54
Have you made an attempt?
– Kavi Rama Murthy
Jul 30 at 9:54
@KaviRamaMurthy Yes I made an attempt to understand why is this true and I read also in the book of Shiryaev but I couldn't figure out.
– Melina
Jul 30 at 9:55
@KaviRamaMurthy Yes I made an attempt to understand why is this true and I read also in the book of Shiryaev but I couldn't figure out.
– Melina
Jul 30 at 9:55
No proof is required. The equality is true by definition.
– Kavi Rama Murthy
Jul 30 at 10:02
No proof is required. The equality is true by definition.
– Kavi Rama Murthy
Jul 30 at 10:02
1
1
Can you find an element in the set denoted on LHS that is not in the set denoted on RHS? Can you find an element in the set denoted on RHS that is not in the set denoted on LHS?
– drhab
Jul 30 at 10:07
Can you find an element in the set denoted on LHS that is not in the set denoted on RHS? Can you find an element in the set denoted on RHS that is not in the set denoted on LHS?
– drhab
Jul 30 at 10:07
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Hint:
If $xin A$ implies that $xin B$ and $xin B$ implies that $xin A$ then we are allowed to conclude that $A=B$.
This because (according to the axiom of extensionality) sets are completely determined by their elements.
answered Jul 30 at 10:11
drhab
85.9k540118
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Hint:
If $xin A$ implies that $xin B$ and $xin B$ implies that $xin A$ then we are allowed to conclude that $A=B$.
This because (according to the axiom of extensionality) sets are completely determined by their elements.
answered Jul 30 at 10:11
drhab
85.9k540118
add a comment |Â
up vote
2
down vote
accepted
Hint:
If $xin A$ implies that $xin B$ and $xin B$ implies that $xin A$ then we are allowed to conclude that $A=B$.
This because (according to the axiom of extensionality) sets are completely determined by their elements.
answered Jul 30 at 10:11
drhab
85.9k540118
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Hint:
If $xin A$ implies that $xin B$ and $xin B$ implies that $xin A$ then we are allowed to conclude that $A=B$.
This because (according to the axiom of extensionality) sets are completely determined by their elements.
answered Jul 30 at 10:11
drhab
85.9k540118
Hint:
If $xin A$ implies that $xin B$ and $xin B$ implies that $xin A$ then we are allowed to conclude that $A=B$.
This because (according to the axiom of extensionality) sets are completely determined by their elements.
answered Jul 30 at 10:11
drhab
85.9k540118
answered Jul 30 at 10:11
drhab
85.9k540118
answered Jul 30 at 10:11
drhab
85.9k540118
answered Jul 30 at 10:11
drhab
85.9k540118
85.9k540118
add a comment |Â
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2866846%2fwhy-a-cylinder-set-with-base-in-mathbbrn-can-have-a-base-in-mathbbrn%23new-answer', 'question_page');
);
Post as a guest
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
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
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
Have you made an attempt?
– Kavi Rama Murthy
Jul 30 at 9:54
@KaviRamaMurthy Yes I made an attempt to understand why is this true and I read also in the book of Shiryaev but I couldn't figure out.
– Melina
Jul 30 at 9:55
No proof is required. The equality is true by definition.
– Kavi Rama Murthy
Jul 30 at 10:02
1
Can you find an element in the set denoted on LHS that is not in the set denoted on RHS? Can you find an element in the set denoted on RHS that is not in the set denoted on LHS?
– drhab
Jul 30 at 10:07