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Why is this set uncountable [closed]

0

$begingroup$

I have an uncountable family $(Omega_i)_{i in L}$ of non empty sets such that

$$
forall i,jin L, ineq j, Omega_icap Omega_j=emptyset$$

and there exists a non empty set $F$ such that $forall iin L,Fcap Omega_i=emptyset.$

How to prove that $bigcup_{iin L} (Fcup Omega_i)$ is uncountable?

thanks

abstract-algebra

share|cite|improve this question

edited Mar 25 at 22:03

Poline Sandra

1208

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

$endgroup$

closed as off-topic by YiFan, Cesareo, Lee David Chung Lin, dantopa, John Omielan Mar 25 at 4:44

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – YiFan, Cesareo, Lee David Chung Lin, dantopa, John Omielan

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Note: $cup_{iin L} (Fcup Omega_i) supseteq cup_{iin L} Omega_i$
  $endgroup$
  – JavaMan
  Mar 24 at 19:23
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  What happens if $Omega_i = emptyset$ for all $i in L$?
  $endgroup$
  – Fabio Somenzi
  Mar 24 at 19:26
  

  
  
  
  

add a comment | 

0

$begingroup$

I have an uncountable family $(Omega_i)_{i in L}$ of non empty sets such that

$$
forall i,jin L, ineq j, Omega_icap Omega_j=emptyset$$

and there exists a non empty set $F$ such that $forall iin L,Fcap Omega_i=emptyset.$

How to prove that $bigcup_{iin L} (Fcup Omega_i)$ is uncountable?

thanks

abstract-algebra

share|cite|improve this question

edited Mar 25 at 22:03

Poline Sandra

1208

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

$endgroup$

closed as off-topic by YiFan, Cesareo, Lee David Chung Lin, dantopa, John Omielan Mar 25 at 4:44

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – YiFan, Cesareo, Lee David Chung Lin, dantopa, John Omielan

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Note: $cup_{iin L} (Fcup Omega_i) supseteq cup_{iin L} Omega_i$
  $endgroup$
  – JavaMan
  Mar 24 at 19:23
  

  
  
  
  


• 
  
  
  

  
  5
  

  
  
  
  
  
  

  
  $begingroup$
  What happens if $Omega_i = emptyset$ for all $i in L$?
  $endgroup$
  – Fabio Somenzi
  Mar 24 at 19:26
  

  
  
  
  

add a comment | 

$begingroup$

I have an uncountable family $(Omega_i)_{i in L}$ of non empty sets such that

$$
forall i,jin L, ineq j, Omega_icap Omega_j=emptyset$$

and there exists a non empty set $F$ such that $forall iin L,Fcap Omega_i=emptyset.$

How to prove that $bigcup_{iin L} (Fcup Omega_i)$ is uncountable?

thanks

abstract-algebra

share|cite|improve this question

edited Mar 25 at 22:03

Poline Sandra

1208

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

$endgroup$

share|cite|improve this question

edited Mar 25 at 22:03

Poline Sandra

1208

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

share|cite|improve this question

edited Mar 25 at 22:03

Poline Sandra

1208

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

edited Mar 25 at 22:03

Poline Sandra

1208

edited Mar 25 at 22:03

Poline Sandra

1208

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

asked Mar 24 at 19:17

VrouvrouVrouvrou

1,9251822

1 Answer
1

active

oldest

votes

1

$begingroup$

$$bigcup_{iin L} (Fcup Omega _i)=Fcupbigcup_{iin L}Omega _i,$$
of course it's uncountable ! Even if $F=emptyset$ (difficult to be smaller), your union is going to be $bigcup_{iin L}Omega _i$, and thus uncountable.

share|cite|improve this answer

edited Mar 25 at 3:58

answered Mar 24 at 19:21

user657324user657324

63310

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  the sets are all non empty
  $endgroup$
  – Vrouvrou
  Mar 24 at 20:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Vrouvrou: Then my answer should help :)
  $endgroup$
  – user657324
  Mar 24 at 20:37
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Vrouvrou It is best to include all relevant details in your question.
  $endgroup$
  – Santana Afton
  Mar 24 at 23:15
  

  
  
  
  

add a comment | 

1

$begingroup$

$$bigcup_{iin L} (Fcup Omega _i)=Fcupbigcup_{iin L}Omega _i,$$
of course it's uncountable ! Even if $F=emptyset$ (difficult to be smaller), your union is going to be $bigcup_{iin L}Omega _i$, and thus uncountable.

share|cite|improve this answer

edited Mar 25 at 3:58

answered Mar 24 at 19:21

user657324user657324

63310

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  the sets are all non empty
  $endgroup$
  – Vrouvrou
  Mar 24 at 20:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Vrouvrou: Then my answer should help :)
  $endgroup$
  – user657324
  Mar 24 at 20:37
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Vrouvrou It is best to include all relevant details in your question.
  $endgroup$
  – Santana Afton
  Mar 24 at 23:15
  

  
  
  
  

add a comment | 

1

$begingroup$

$$bigcup_{iin L} (Fcup Omega _i)=Fcupbigcup_{iin L}Omega _i,$$
of course it's uncountable ! Even if $F=emptyset$ (difficult to be smaller), your union is going to be $bigcup_{iin L}Omega _i$, and thus uncountable.

share|cite|improve this answer

edited Mar 25 at 3:58

answered Mar 24 at 19:21

user657324user657324

63310

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  the sets are all non empty
  $endgroup$
  – Vrouvrou
  Mar 24 at 20:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Vrouvrou: Then my answer should help :)
  $endgroup$
  – user657324
  Mar 24 at 20:37
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Vrouvrou It is best to include all relevant details in your question.
  $endgroup$
  – Santana Afton
  Mar 24 at 23:15
  

  
  
  
  

add a comment | 

$begingroup$

$$bigcup_{iin L} (Fcup Omega _i)=Fcupbigcup_{iin L}Omega _i,$$
of course it's uncountable ! Even if $F=emptyset$ (difficult to be smaller), your union is going to be $bigcup_{iin L}Omega _i$, and thus uncountable.

share|cite|improve this answer

edited Mar 25 at 3:58

answered Mar 24 at 19:21

user657324user657324

63310

$endgroup$

share|cite|improve this answer

edited Mar 25 at 3:58

answered Mar 24 at 19:21

user657324user657324

63310

answered Mar 24 at 19:21

user657324user657324

63310

answered Mar 24 at 19:21

user657324user657324

63310