Why is it called a “family†of sets?
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If I'm not mistaken, a "family of sets" is just $f:Imapsto X$. My question is: why they decided to call it a "family" of sets? What's the intuitive picture that made some guy call this a $textbffamily$?
elementary-set-theory terminology
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
asked Aug 18 at 3:28
Ali
916
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up vote
-2
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If I'm not mistaken, a "family of sets" is just $f:Imapsto X$. My question is: why they decided to call it a "family" of sets? What's the intuitive picture that made some guy call this a $textbffamily$?
elementary-set-theory terminology
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
asked Aug 18 at 3:28
Ali
916
2
What do you think is X in your description? Family is a lot like collection, group or set, but it doesn’t have the same connotations necessarily. You generally try to pick terms that way.
– rschwieb
Aug 18 at 3:33
1
Maybe someone got tired of saying "set of sets". Seriously, though, it's easier to keep your thoughts organized when you give different kinds of names. like family, or collection, etc., to some sets. Intuitively the members of a family of sets ought to be related, in that they are all the sets with some specified property.
– DanielWainfleet
Aug 18 at 5:14
Generally a family of sets is the same thing as a set of sets. And in pure set-theory it's redundant because everything that exists is a set.
– DanielWainfleet
Aug 18 at 5:21
1
Often the sets in the family are related in some way. They are not just a random collection of sets that are the image of $f$. The powerset of some set $A$ is a family-they are all subsets of $A$. They might be partial functions from $omega$ to $0,1$.. They might be lots of things.
– Ross Millikan
Aug 18 at 5:29
Thanks all. Especially @RossMillikan. It was exactly the explanation I was looking for.
– Ali
Aug 18 at 7:13
 |Â
show 1 more comment
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
If I'm not mistaken, a "family of sets" is just $f:Imapsto X$. My question is: why they decided to call it a "family" of sets? What's the intuitive picture that made some guy call this a $textbffamily$?
elementary-set-theory terminology
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
asked Aug 18 at 3:28
Ali
916
If I'm not mistaken, a "family of sets" is just $f:Imapsto X$. My question is: why they decided to call it a "family" of sets? What's the intuitive picture that made some guy call this a $textbffamily$?
elementary-set-theory terminology
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
asked Aug 18 at 3:28
Ali
916
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
edited Aug 18 at 7:27
Asaf Karagila♦
293k31407735
293k31407735
asked Aug 18 at 3:28
Ali
916
asked Aug 18 at 3:28
Ali
916
asked Aug 18 at 3:28
Ali
916
916
2
What do you think is X in your description? Family is a lot like collection, group or set, but it doesn’t have the same connotations necessarily. You generally try to pick terms that way.
– rschwieb
Aug 18 at 3:33
1
Maybe someone got tired of saying "set of sets". Seriously, though, it's easier to keep your thoughts organized when you give different kinds of names. like family, or collection, etc., to some sets. Intuitively the members of a family of sets ought to be related, in that they are all the sets with some specified property.
– DanielWainfleet
Aug 18 at 5:14
Generally a family of sets is the same thing as a set of sets. And in pure set-theory it's redundant because everything that exists is a set.
– DanielWainfleet
Aug 18 at 5:21
1
Often the sets in the family are related in some way. They are not just a random collection of sets that are the image of $f$. The powerset of some set $A$ is a family-they are all subsets of $A$. They might be partial functions from $omega$ to $0,1$.. They might be lots of things.
– Ross Millikan
Aug 18 at 5:29
Thanks all. Especially @RossMillikan. It was exactly the explanation I was looking for.
– Ali
Aug 18 at 7:13
 |Â
show 1 more comment
2
What do you think is X in your description? Family is a lot like collection, group or set, but it doesn’t have the same connotations necessarily. You generally try to pick terms that way.
– rschwieb
Aug 18 at 3:33
1
Maybe someone got tired of saying "set of sets". Seriously, though, it's easier to keep your thoughts organized when you give different kinds of names. like family, or collection, etc., to some sets. Intuitively the members of a family of sets ought to be related, in that they are all the sets with some specified property.
– DanielWainfleet
Aug 18 at 5:14
Generally a family of sets is the same thing as a set of sets. And in pure set-theory it's redundant because everything that exists is a set.
– DanielWainfleet
Aug 18 at 5:21
1
Often the sets in the family are related in some way. They are not just a random collection of sets that are the image of $f$. The powerset of some set $A$ is a family-they are all subsets of $A$. They might be partial functions from $omega$ to $0,1$.. They might be lots of things.
– Ross Millikan
Aug 18 at 5:29
Thanks all. Especially @RossMillikan. It was exactly the explanation I was looking for.
– Ali
Aug 18 at 7:13
2
2
What do you think is X in your description? Family is a lot like collection, group or set, but it doesn’t have the same connotations necessarily. You generally try to pick terms that way.
– rschwieb
Aug 18 at 3:33
What do you think is X in your description? Family is a lot like collection, group or set, but it doesn’t have the same connotations necessarily. You generally try to pick terms that way.
– rschwieb
Aug 18 at 3:33
1
1
Maybe someone got tired of saying "set of sets". Seriously, though, it's easier to keep your thoughts organized when you give different kinds of names. like family, or collection, etc., to some sets. Intuitively the members of a family of sets ought to be related, in that they are all the sets with some specified property.
– DanielWainfleet
Aug 18 at 5:14
Maybe someone got tired of saying "set of sets". Seriously, though, it's easier to keep your thoughts organized when you give different kinds of names. like family, or collection, etc., to some sets. Intuitively the members of a family of sets ought to be related, in that they are all the sets with some specified property.
– DanielWainfleet
Aug 18 at 5:14
Generally a family of sets is the same thing as a set of sets. And in pure set-theory it's redundant because everything that exists is a set.
– DanielWainfleet
Aug 18 at 5:21
Generally a family of sets is the same thing as a set of sets. And in pure set-theory it's redundant because everything that exists is a set.
– DanielWainfleet
Aug 18 at 5:21
1
1
Often the sets in the family are related in some way. They are not just a random collection of sets that are the image of $f$. The powerset of some set $A$ is a family-they are all subsets of $A$. They might be partial functions from $omega$ to $0,1$.. They might be lots of things.
– Ross Millikan
Aug 18 at 5:29
Often the sets in the family are related in some way. They are not just a random collection of sets that are the image of $f$. The powerset of some set $A$ is a family-they are all subsets of $A$. They might be partial functions from $omega$ to $0,1$.. They might be lots of things.
– Ross Millikan
Aug 18 at 5:29
Thanks all. Especially @RossMillikan. It was exactly the explanation I was looking for.
– Ali
Aug 18 at 7:13
Thanks all. Especially @RossMillikan. It was exactly the explanation I was looking for.
– Ali
Aug 18 at 7:13
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
A family of things is certainly different from a set of things.
If $A$ is a nonempty set then its elements $x$ are not "organized" in any way. Each thing $x$ you can think of either is an element of $A$, or is not an element of $A$, period. Given $A$ you can, e.g., form the set $Achoose 3$of all three-element subsets of $A$. This is a set of sets, but not a family of sets.
In order to obtain a family of things $x$ you need an index set $I$ and a function $f$ that produces for each $iotain I$ one such thing $x_iota:=f(iota)$. The resulting "data structure" is then called a family of things, and is denoted by $bigl(x_iotabigr)_iotain I$.
As an example, if $X$ is an arbitrary nonempty set and $cal P(X)$ its power set then an $$f:>Itocal P(X),quadiotato A_iota$$ produces a family of subsets of $X$. This family is denoted by $bigl(A_iotabigr)_iotain I$. Note that the same subset $Asubset X$ may occur several times in this family, whereas in a set of subsets it would occur at most once.
The functions $f$ appearing in this explanation are not interesting per se; therefore they do not appear in the notation. But every time we talk about a "family" such an $f$ is looming in the background.
answered Aug 18 at 14:34
Christian Blatter
165k7109309
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
1
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
A family of things is certainly different from a set of things.
If $A$ is a nonempty set then its elements $x$ are not "organized" in any way. Each thing $x$ you can think of either is an element of $A$, or is not an element of $A$, period. Given $A$ you can, e.g., form the set $Achoose 3$of all three-element subsets of $A$. This is a set of sets, but not a family of sets.
In order to obtain a family of things $x$ you need an index set $I$ and a function $f$ that produces for each $iotain I$ one such thing $x_iota:=f(iota)$. The resulting "data structure" is then called a family of things, and is denoted by $bigl(x_iotabigr)_iotain I$.
As an example, if $X$ is an arbitrary nonempty set and $cal P(X)$ its power set then an $$f:>Itocal P(X),quadiotato A_iota$$ produces a family of subsets of $X$. This family is denoted by $bigl(A_iotabigr)_iotain I$. Note that the same subset $Asubset X$ may occur several times in this family, whereas in a set of subsets it would occur at most once.
The functions $f$ appearing in this explanation are not interesting per se; therefore they do not appear in the notation. But every time we talk about a "family" such an $f$ is looming in the background.
answered Aug 18 at 14:34
Christian Blatter
165k7109309
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
1
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
add a comment |Â
up vote
1
down vote
accepted
A family of things is certainly different from a set of things.
If $A$ is a nonempty set then its elements $x$ are not "organized" in any way. Each thing $x$ you can think of either is an element of $A$, or is not an element of $A$, period. Given $A$ you can, e.g., form the set $Achoose 3$of all three-element subsets of $A$. This is a set of sets, but not a family of sets.
In order to obtain a family of things $x$ you need an index set $I$ and a function $f$ that produces for each $iotain I$ one such thing $x_iota:=f(iota)$. The resulting "data structure" is then called a family of things, and is denoted by $bigl(x_iotabigr)_iotain I$.
As an example, if $X$ is an arbitrary nonempty set and $cal P(X)$ its power set then an $$f:>Itocal P(X),quadiotato A_iota$$ produces a family of subsets of $X$. This family is denoted by $bigl(A_iotabigr)_iotain I$. Note that the same subset $Asubset X$ may occur several times in this family, whereas in a set of subsets it would occur at most once.
The functions $f$ appearing in this explanation are not interesting per se; therefore they do not appear in the notation. But every time we talk about a "family" such an $f$ is looming in the background.
answered Aug 18 at 14:34
Christian Blatter
165k7109309
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
1
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
A family of things is certainly different from a set of things.
If $A$ is a nonempty set then its elements $x$ are not "organized" in any way. Each thing $x$ you can think of either is an element of $A$, or is not an element of $A$, period. Given $A$ you can, e.g., form the set $Achoose 3$of all three-element subsets of $A$. This is a set of sets, but not a family of sets.
In order to obtain a family of things $x$ you need an index set $I$ and a function $f$ that produces for each $iotain I$ one such thing $x_iota:=f(iota)$. The resulting "data structure" is then called a family of things, and is denoted by $bigl(x_iotabigr)_iotain I$.
As an example, if $X$ is an arbitrary nonempty set and $cal P(X)$ its power set then an $$f:>Itocal P(X),quadiotato A_iota$$ produces a family of subsets of $X$. This family is denoted by $bigl(A_iotabigr)_iotain I$. Note that the same subset $Asubset X$ may occur several times in this family, whereas in a set of subsets it would occur at most once.
The functions $f$ appearing in this explanation are not interesting per se; therefore they do not appear in the notation. But every time we talk about a "family" such an $f$ is looming in the background.
answered Aug 18 at 14:34
Christian Blatter
165k7109309
A family of things is certainly different from a set of things.
If $A$ is a nonempty set then its elements $x$ are not "organized" in any way. Each thing $x$ you can think of either is an element of $A$, or is not an element of $A$, period. Given $A$ you can, e.g., form the set $Achoose 3$of all three-element subsets of $A$. This is a set of sets, but not a family of sets.
In order to obtain a family of things $x$ you need an index set $I$ and a function $f$ that produces for each $iotain I$ one such thing $x_iota:=f(iota)$. The resulting "data structure" is then called a family of things, and is denoted by $bigl(x_iotabigr)_iotain I$.
As an example, if $X$ is an arbitrary nonempty set and $cal P(X)$ its power set then an $$f:>Itocal P(X),quadiotato A_iota$$ produces a family of subsets of $X$. This family is denoted by $bigl(A_iotabigr)_iotain I$. Note that the same subset $Asubset X$ may occur several times in this family, whereas in a set of subsets it would occur at most once.
The functions $f$ appearing in this explanation are not interesting per se; therefore they do not appear in the notation. But every time we talk about a "family" such an $f$ is looming in the background.
answered Aug 18 at 14:34
Christian Blatter
165k7109309
answered Aug 18 at 14:34
Christian Blatter
165k7109309
answered Aug 18 at 14:34
Christian Blatter
165k7109309
answered Aug 18 at 14:34
Christian Blatter
165k7109309
165k7109309
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
1
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
add a comment |Â
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
1
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
Perfect, this explanation is compatible with the book I'm reading (Bourbaki's set theory)
– Ali
Aug 18 at 16:43
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
One quick question: when you say "the same subset may occur several times", is it the same as saying "several members of $I$ map to the same element in $cal P(X)$"?
– Ali
Aug 18 at 16:54
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
Also, is it correct to say that a "family of sets" is exactly the same as a mapping but the terminology is used when the codomain is a set of sets? Thanks.
– Ali
Aug 18 at 17:00
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
@Ali: Why on earth are you reading that book?
– Asaf Karagila♦
Aug 18 at 18:09
1
1
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
Concerning your first question: Yes. The $n$-tuple $(5,1,-2,1,4)$ is a family of integers, indexed by the set $1,2,3,4,5$. Concerning your second question: We use the word "family" instead of "mapping" when we are not interested in the exact $f$ and its properties, but in the "list" of produced objects.
– Christian Blatter
Aug 18 at 19:06
add a comment |Â
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2
What do you think is X in your description? Family is a lot like collection, group or set, but it doesn’t have the same connotations necessarily. You generally try to pick terms that way.
– rschwieb
Aug 18 at 3:33
1
Maybe someone got tired of saying "set of sets". Seriously, though, it's easier to keep your thoughts organized when you give different kinds of names. like family, or collection, etc., to some sets. Intuitively the members of a family of sets ought to be related, in that they are all the sets with some specified property.
– DanielWainfleet
Aug 18 at 5:14
Generally a family of sets is the same thing as a set of sets. And in pure set-theory it's redundant because everything that exists is a set.
– DanielWainfleet
Aug 18 at 5:21
1
Often the sets in the family are related in some way. They are not just a random collection of sets that are the image of $f$. The powerset of some set $A$ is a family-they are all subsets of $A$. They might be partial functions from $omega$ to $0,1$.. They might be lots of things.
– Ross Millikan
Aug 18 at 5:29
Thanks all. Especially @RossMillikan. It was exactly the explanation I was looking for.
– Ali
Aug 18 at 7:13