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What is a furoni family?

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0

$begingroup$

What exactly is a Furoni family please explain with relevant examples?
This contest paper describes Furoni family as

We call
a set, $F$, of subsets of $C_n = 1,2,dots,n$ a Furoni family of $C_n$ if no element of $F$ is a subset of
another element of $F$.

For example, one question in the given contest paper is:

"(a) Consider $A = 1,2, 1,3, 1,4 $. Note that $A$ is a Furoni family of $C_4$. Determine the two Furoni families of $C_4$ that contain all of the elements of $A$ and to which no other subsets of $C_4$ can be added to form a new (larger) Furoni family."

Please explain with examples.

number-theory contest-math

share|cite|improve this question

edited yesterday

Santana Afton

3,0182630

asked yesterday

Kartik.kKartik.k

91

New contributor

Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Hi, Kartik.k! Welcome to Math.SE. In order to get good quality answers, it is best to include as much context as possible. What is "the contest paper"? What is $Cn$? etc.
  $endgroup$
  – Santana Afton
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  en.wikipedia.org/wiki/Sperner_family
  $endgroup$
  – vadim123
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What don't you understand about the given definition?
  $endgroup$
  – Eric Wofsey
  21 hours ago
  

  
  
  
  

add a comment | 

0

$begingroup$

What exactly is a Furoni family please explain with relevant examples?
This contest paper describes Furoni family as

We call
a set, $F$, of subsets of $C_n = 1,2,dots,n$ a Furoni family of $C_n$ if no element of $F$ is a subset of
another element of $F$.

For example, one question in the given contest paper is:

"(a) Consider $A = 1,2, 1,3, 1,4 $. Note that $A$ is a Furoni family of $C_4$. Determine the two Furoni families of $C_4$ that contain all of the elements of $A$ and to which no other subsets of $C_4$ can be added to form a new (larger) Furoni family."

Please explain with examples.

number-theory contest-math

share|cite|improve this question

edited yesterday

Santana Afton

3,0182630

asked yesterday

Kartik.kKartik.k

91

New contributor

Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Hi, Kartik.k! Welcome to Math.SE. In order to get good quality answers, it is best to include as much context as possible. What is "the contest paper"? What is $Cn$? etc.
  $endgroup$
  – Santana Afton
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  en.wikipedia.org/wiki/Sperner_family
  $endgroup$
  – vadim123
  yesterday
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What don't you understand about the given definition?
  $endgroup$
  – Eric Wofsey
  21 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

What exactly is a Furoni family please explain with relevant examples?
This contest paper describes Furoni family as

We call
a set, $F$, of subsets of $C_n = 1,2,dots,n$ a Furoni family of $C_n$ if no element of $F$ is a subset of
another element of $F$.

For example, one question in the given contest paper is:

"(a) Consider $A = 1,2, 1,3, 1,4 $. Note that $A$ is a Furoni family of $C_4$. Determine the two Furoni families of $C_4$ that contain all of the elements of $A$ and to which no other subsets of $C_4$ can be added to form a new (larger) Furoni family."

Please explain with examples.

number-theory contest-math

share|cite|improve this question

edited yesterday

Santana Afton

3,0182630

asked yesterday

Kartik.kKartik.k

91

New contributor

Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited yesterday

Santana Afton

3,0182630

asked yesterday

Kartik.kKartik.k

91

New contributor

Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited yesterday

Santana Afton

3,0182630

asked yesterday

Kartik.kKartik.k

91

New contributor

Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited yesterday

Santana Afton

3,0182630

edited yesterday

Santana Afton

3,0182630

asked yesterday

Kartik.kKartik.k

91

New contributor

Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

Kartik.kKartik.k

91

1 Answer
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$begingroup$

According to your definition of Furoni family, if we define $$C_3=1,2,3$$then $$F=Big1,2,3Big$$is a Furoni family of $C_3$, since no element of $F$ is a subset of any other element of $F$, i.e.$$1notsubset2,3\2,3notsubset1$$

Edit

To answer the new added-up question, note that $C_4$ is provided by $16$ different subsets. In the case of this question, the empty set ($emptyset$) can't be added to the Furoni family since it is a subset of every set. Two requested Furoni families then go like this:$$F_1=Big1,2,3,4Big\F_2=Big1,2,3,4Big$$For $F_1$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $ksubset S$ for some $k=1,2,3,4$.

Similarly for $F_2$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $Ssubset1,2,3,4$.

share|cite|improve this answer

edited 22 hours ago

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks. Please answer this question. "(a) Consider A=1,2,1,3,1,4. Note that A is a Furoni family of C4. Determine the two Furoni families of C4 that contain all of the elements of A and to which no other subsets of C4 can be added to form a new (larger) Furoni family."
  $endgroup$
  – Kartik.k
  23 hours ago
  
  

  
  
  
  

add a comment | 

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2

$begingroup$

According to your definition of Furoni family, if we define $$C_3=1,2,3$$then $$F=Big1,2,3Big$$is a Furoni family of $C_3$, since no element of $F$ is a subset of any other element of $F$, i.e.$$1notsubset2,3\2,3notsubset1$$

Edit

To answer the new added-up question, note that $C_4$ is provided by $16$ different subsets. In the case of this question, the empty set ($emptyset$) can't be added to the Furoni family since it is a subset of every set. Two requested Furoni families then go like this:$$F_1=Big1,2,3,4Big\F_2=Big1,2,3,4Big$$For $F_1$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $ksubset S$ for some $k=1,2,3,4$.

Similarly for $F_2$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $Ssubset1,2,3,4$.

share|cite|improve this answer

edited 22 hours ago

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks. Please answer this question. "(a) Consider A=1,2,1,3,1,4. Note that A is a Furoni family of C4. Determine the two Furoni families of C4 that contain all of the elements of A and to which no other subsets of C4 can be added to form a new (larger) Furoni family."
  $endgroup$
  – Kartik.k
  23 hours ago
  
  

  
  
  
  

add a comment | 

2

$begingroup$

According to your definition of Furoni family, if we define $$C_3=1,2,3$$then $$F=Big1,2,3Big$$is a Furoni family of $C_3$, since no element of $F$ is a subset of any other element of $F$, i.e.$$1notsubset2,3\2,3notsubset1$$

Edit

To answer the new added-up question, note that $C_4$ is provided by $16$ different subsets. In the case of this question, the empty set ($emptyset$) can't be added to the Furoni family since it is a subset of every set. Two requested Furoni families then go like this:$$F_1=Big1,2,3,4Big\F_2=Big1,2,3,4Big$$For $F_1$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $ksubset S$ for some $k=1,2,3,4$.

Similarly for $F_2$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $Ssubset1,2,3,4$.

share|cite|improve this answer

edited 22 hours ago

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks. Please answer this question. "(a) Consider A=1,2,1,3,1,4. Note that A is a Furoni family of C4. Determine the two Furoni families of C4 that contain all of the elements of A and to which no other subsets of C4 can be added to form a new (larger) Furoni family."
  $endgroup$
  – Kartik.k
  23 hours ago
  
  

  
  
  
  

add a comment | 

$begingroup$

According to your definition of Furoni family, if we define $$C_3=1,2,3$$then $$F=Big1,2,3Big$$is a Furoni family of $C_3$, since no element of $F$ is a subset of any other element of $F$, i.e.$$1notsubset2,3\2,3notsubset1$$

Edit

To answer the new added-up question, note that $C_4$ is provided by $16$ different subsets. In the case of this question, the empty set ($emptyset$) can't be added to the Furoni family since it is a subset of every set. Two requested Furoni families then go like this:$$F_1=Big1,2,3,4Big\F_2=Big1,2,3,4Big$$For $F_1$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $ksubset S$ for some $k=1,2,3,4$.

Similarly for $F_2$ if any other subset (say $Ssubseteq1,2,3,4$) is added then $Ssubset1,2,3,4$.

share|cite|improve this answer

edited 22 hours ago

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

$endgroup$

share|cite|improve this answer

edited 22 hours ago

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040

answered yesterday

Mostafa AyazMostafa Ayaz

18.1k31040