Intersection of sets of subsets

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What is the definition of an intersection of sets of subsets?

Say, I have a set $X$ and its subsets $A,B$. I form 2 sets of subsets:

$X_1=X,emptyset,A,Xsetminus A\
X_2=X,emptyset,B,Xsetminus B$,

what is $X_1cap X_2$? Is the definition based on subsets (that both sets of subsets have to contain a particular subset --- in the case above it would only be $X,emptyset$) or based on elements (I think then it should be:
$X_1cap X_2 =(xin Xlor xin A lor xin Xsetminus Alor x =emptyset)land\
(xin Xlor xin B lor xin Xsetminus Blor x =emptyset)$
)

measure-theory elementary-set-theory

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edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

asked Jul 18 at 18:07

leosenko

1778

• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be just $X, emptyset$ assuming that $A neq B$ or $A cup B neq X$
  – Anurag A
  Jul 18 at 18:08
  
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

What is the definition of an intersection of sets of subsets?

Say, I have a set $X$ and its subsets $A,B$. I form 2 sets of subsets:

$X_1=X,emptyset,A,Xsetminus A\
X_2=X,emptyset,B,Xsetminus B$,

what is $X_1cap X_2$? Is the definition based on subsets (that both sets of subsets have to contain a particular subset --- in the case above it would only be $X,emptyset$) or based on elements (I think then it should be:
$X_1cap X_2 =(xin Xlor xin A lor xin Xsetminus Alor x =emptyset)land\
(xin Xlor xin B lor xin Xsetminus Blor x =emptyset)$
)

measure-theory elementary-set-theory

share|cite|improve this question

edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

asked Jul 18 at 18:07

leosenko

1778

• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be just $X, emptyset$ assuming that $A neq B$ or $A cup B neq X$
  – Anurag A
  Jul 18 at 18:08
  
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

What is the definition of an intersection of sets of subsets?

Say, I have a set $X$ and its subsets $A,B$. I form 2 sets of subsets:

$X_1=X,emptyset,A,Xsetminus A\
X_2=X,emptyset,B,Xsetminus B$,

what is $X_1cap X_2$? Is the definition based on subsets (that both sets of subsets have to contain a particular subset --- in the case above it would only be $X,emptyset$) or based on elements (I think then it should be:
$X_1cap X_2 =(xin Xlor xin A lor xin Xsetminus Alor x =emptyset)land\
(xin Xlor xin B lor xin Xsetminus Blor x =emptyset)$
)

measure-theory elementary-set-theory

share|cite|improve this question

edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

asked Jul 18 at 18:07

leosenko

1778

What is the definition of an intersection of sets of subsets?

Say, I have a set $X$ and its subsets $A,B$. I form 2 sets of subsets:

$X_1=X,emptyset,A,Xsetminus A\
X_2=X,emptyset,B,Xsetminus B$,

what is $X_1cap X_2$? Is the definition based on subsets (that both sets of subsets have to contain a particular subset --- in the case above it would only be $X,emptyset$) or based on elements (I think then it should be:
$X_1cap X_2 =(xin Xlor xin A lor xin Xsetminus Alor x =emptyset)land\
(xin Xlor xin B lor xin Xsetminus Blor x =emptyset)$
)

measure-theory elementary-set-theory

share|cite|improve this question

edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

asked Jul 18 at 18:07

leosenko

1778

share|cite|improve this question

share|cite|improve this question

edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

edited Jul 18 at 20:27

Andrés E. Caicedo

63.2k7151235

63.2k7151235

asked Jul 18 at 18:07

leosenko

1778

asked Jul 18 at 18:07

leosenko

1778

asked Jul 18 at 18:07

leosenko

1778

1778

• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be just $X, emptyset$ assuming that $A neq B$ or $A cup B neq X$
  – Anurag A
  Jul 18 at 18:08
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be just $X, emptyset$ assuming that $A neq B$ or $A cup B neq X$
  – Anurag A
  Jul 18 at 18:08
  
  

  
  
  
  

It should be just $X, emptyset$ assuming that $A neq B$ or $A cup B neq X$
– Anurag A
Jul 18 at 18:08

It should be just $X, emptyset$ assuming that $A neq B$ or $A cup B neq X$
– Anurag A
Jul 18 at 18:08

add a comment | 

1 Answer
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The notation $X_1cap X_2$ always refers to the intersection of the sets $X_1$ and $X_2$. That is, $$X_1cap X_2=x:xin X_1wedge xin X_2.$$ In your case, that would be $X,emptyset$ (at least as long as $Aneq B$ and $Aneq Xsetminus B$).

(There are situations where $X_1*X_2$ can refer to the set $x_1*x_2:x_1in X_1,x_2in X_2$ where $*$ is some binary operation. I've never see this usage when $*$ is intersection, though, and certainly no author would do so for intersection without commenting on what they are doing.)

share|cite|improve this answer

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

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

up vote
3
down vote

The notation $X_1cap X_2$ always refers to the intersection of the sets $X_1$ and $X_2$. That is, $$X_1cap X_2=x:xin X_1wedge xin X_2.$$ In your case, that would be $X,emptyset$ (at least as long as $Aneq B$ and $Aneq Xsetminus B$).

(There are situations where $X_1*X_2$ can refer to the set $x_1*x_2:x_1in X_1,x_2in X_2$ where $*$ is some binary operation. I've never see this usage when $*$ is intersection, though, and certainly no author would do so for intersection without commenting on what they are doing.)

share|cite|improve this answer

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

add a comment | 

up vote
3
down vote

The notation $X_1cap X_2$ always refers to the intersection of the sets $X_1$ and $X_2$. That is, $$X_1cap X_2=x:xin X_1wedge xin X_2.$$ In your case, that would be $X,emptyset$ (at least as long as $Aneq B$ and $Aneq Xsetminus B$).

(There are situations where $X_1*X_2$ can refer to the set $x_1*x_2:x_1in X_1,x_2in X_2$ where $*$ is some binary operation. I've never see this usage when $*$ is intersection, though, and certainly no author would do so for intersection without commenting on what they are doing.)

share|cite|improve this answer

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

add a comment | 

up vote
3
down vote

up vote
3
down vote

The notation $X_1cap X_2$ always refers to the intersection of the sets $X_1$ and $X_2$. That is, $$X_1cap X_2=x:xin X_1wedge xin X_2.$$ In your case, that would be $X,emptyset$ (at least as long as $Aneq B$ and $Aneq Xsetminus B$).

(There are situations where $X_1*X_2$ can refer to the set $x_1*x_2:x_1in X_1,x_2in X_2$ where $*$ is some binary operation. I've never see this usage when $*$ is intersection, though, and certainly no author would do so for intersection without commenting on what they are doing.)

share|cite|improve this answer

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

The notation $X_1cap X_2$ always refers to the intersection of the sets $X_1$ and $X_2$. That is, $$X_1cap X_2=x:xin X_1wedge xin X_2.$$ In your case, that would be $X,emptyset$ (at least as long as $Aneq B$ and $Aneq Xsetminus B$).

(There are situations where $X_1*X_2$ can refer to the set $x_1*x_2:x_1in X_1,x_2in X_2$ where $*$ is some binary operation. I've never see this usage when $*$ is intersection, though, and certainly no author would do so for intersection without commenting on what they are doing.)

share|cite|improve this answer

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

share|cite|improve this answer

share|cite|improve this answer

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

answered Jul 18 at 18:11

Eric Wofsey

162k12189300

162k12189300

add a comment | 

add a comment | 

 

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