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What is a furoni family?
Producing infinite family of transcendental numbersEquation with divisors IIFamily of elliptic curves with trivial torsionWhat are some of the more efficient ways of studying for an Olympiad?Find the number of members of a familyIndistinguishable pairs, distinguishable triples of metal circles in key-ring jumble.What is the motivation behind the solution of this olympiad problem?What is the least positive integer $x$ such that $x^2$ starts with 2017?A smaller Sidel'nikov sequence is embedded into a longer one. When and how?Family of $A^2=I$
$begingroup$
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
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.
$endgroup$
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
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
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.
$endgroup$
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
number-theory contest-math
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
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
edited yesterday
Santana Afton
3,0182630
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
asked yesterday
Kartik.kKartik.k
91
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.
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.
Kartik.k is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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 |
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
1
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
$begingroup$
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
$begingroup$
en.wikipedia.org/wiki/Sperner_family
$endgroup$
– vadim123
yesterday
1
1
$begingroup$
What don't you understand about the given definition?
$endgroup$
– Eric Wofsey
21 hours ago
$begingroup$
What don't you understand about the given definition?
$endgroup$
– Eric Wofsey
21 hours ago
add a comment |
1 Answer
1
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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 $Ssubset1,2,3,4$.
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 |
Your 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 $Ssubset1,2,3,4$.
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 $Ssubset1,2,3,4$.
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 $Ssubset1,2,3,4$.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
$endgroup$
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 $Ssubset1,2,3,4$.
answered yesterday
Mostafa AyazMostafa Ayaz
18.1k31040
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
18.1k31040
$begingroup$
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 |
$begingroup$
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
$begingroup$
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
$begingroup$
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 |
Kartik.k is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
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