Mixing
Symmetric group and the empty set
Clash Royale CLAN TAG#URR8PPP
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I have a set $X=left 1,...,nright $ and the symmetric group on a set of n elements has order n!.
When $n=0$,why do we have $S_left 1,...,0right =S_varnothing $ ?
I know that n=0 means there are no elements in the set and $0!=1$ but why do we have
$left 1,...,0right =varnothing $? Is it not simply $varnothing =left right $?
Thank you.
abstract-algebra group-theory elementary-set-theory
asked Jul 16 at 0:21
user159729
1
add a comment |Â
up vote
-1
down vote
favorite
I have a set $X=left 1,...,nright $ and the symmetric group on a set of n elements has order n!.
When $n=0$,why do we have $S_left 1,...,0right =S_varnothing $ ?
I know that n=0 means there are no elements in the set and $0!=1$ but why do we have
$left 1,...,0right =varnothing $? Is it not simply $varnothing =left right $?
Thank you.
abstract-algebra group-theory elementary-set-theory
asked Jul 16 at 0:21
user159729
1
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I have a set $X=left 1,...,nright $ and the symmetric group on a set of n elements has order n!.
When $n=0$,why do we have $S_left 1,...,0right =S_varnothing $ ?
I know that n=0 means there are no elements in the set and $0!=1$ but why do we have
$left 1,...,0right =varnothing $? Is it not simply $varnothing =left right $?
Thank you.
abstract-algebra group-theory elementary-set-theory
asked Jul 16 at 0:21
user159729
1
I have a set $X=left 1,...,nright $ and the symmetric group on a set of n elements has order n!.
When $n=0$,why do we have $S_left 1,...,0right =S_varnothing $ ?
I know that n=0 means there are no elements in the set and $0!=1$ but why do we have
$left 1,...,0right =varnothing $? Is it not simply $varnothing =left right $?
Thank you.
abstract-algebra group-theory elementary-set-theory
asked Jul 16 at 0:21
user159729
1
asked Jul 16 at 0:21
user159729
1
asked Jul 16 at 0:21
user159729
1
asked Jul 16 at 0:21
user159729
1
1
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
0
down vote
Your $X$ makes no sense when $n=0$ so it must be defined separately (if you really wanted to, but I don’t know anyone who cares about $S_0$). The natural thing is for $S_0$ to stand for the group of all bijections on the set with no object, the empty set. There is exactly one such function (the empty function) so $S_0$ is trivial, just like $S_1$. And, $0!=1!=1$ so everything is coherent.
answered Jul 16 at 0:50
Randall
7,2471825
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
add a comment |Â
up vote
0
down vote
Think of $X = 1,...,n$ as being shorthand for $X = i in mathbbZ mid 1 le i le n$.
So if $n=0$ then $X=emptyset$.
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
1
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Your $X$ makes no sense when $n=0$ so it must be defined separately (if you really wanted to, but I don’t know anyone who cares about $S_0$). The natural thing is for $S_0$ to stand for the group of all bijections on the set with no object, the empty set. There is exactly one such function (the empty function) so $S_0$ is trivial, just like $S_1$. And, $0!=1!=1$ so everything is coherent.
answered Jul 16 at 0:50
Randall
7,2471825
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
add a comment |Â
up vote
0
down vote
Your $X$ makes no sense when $n=0$ so it must be defined separately (if you really wanted to, but I don’t know anyone who cares about $S_0$). The natural thing is for $S_0$ to stand for the group of all bijections on the set with no object, the empty set. There is exactly one such function (the empty function) so $S_0$ is trivial, just like $S_1$. And, $0!=1!=1$ so everything is coherent.
answered Jul 16 at 0:50
Randall
7,2471825
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Your $X$ makes no sense when $n=0$ so it must be defined separately (if you really wanted to, but I don’t know anyone who cares about $S_0$). The natural thing is for $S_0$ to stand for the group of all bijections on the set with no object, the empty set. There is exactly one such function (the empty function) so $S_0$ is trivial, just like $S_1$. And, $0!=1!=1$ so everything is coherent.
answered Jul 16 at 0:50
Randall
7,2471825
Your $X$ makes no sense when $n=0$ so it must be defined separately (if you really wanted to, but I don’t know anyone who cares about $S_0$). The natural thing is for $S_0$ to stand for the group of all bijections on the set with no object, the empty set. There is exactly one such function (the empty function) so $S_0$ is trivial, just like $S_1$. And, $0!=1!=1$ so everything is coherent.
answered Jul 16 at 0:50
Randall
7,2471825
edited Jul 16 at 1:18
answered Jul 16 at 0:50
Randall
7,2471825
answered Jul 16 at 0:50
Randall
7,2471825
answered Jul 16 at 0:50
Randall
7,2471825
7,2471825
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
add a comment |Â
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I was wondering what the teacher meant when they wrote $S_left 1,...,0right =S_varnothing $, it is only a shortened notation then I guess.I thought it was strange.Thanks again.
– user159729
Jul 16 at 1:11
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
I wouldn’t have written that myself because it looks either wrong or weird or both. Don’t tell your teacher I said that.
– Randall
Jul 16 at 1:12
add a comment |Â
up vote
0
down vote
Think of $X = 1,...,n$ as being shorthand for $X = i in mathbbZ mid 1 le i le n$.
So if $n=0$ then $X=emptyset$.
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
1
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
add a comment |Â
up vote
0
down vote
Think of $X = 1,...,n$ as being shorthand for $X = i in mathbbZ mid 1 le i le n$.
So if $n=0$ then $X=emptyset$.
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
1
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Think of $X = 1,...,n$ as being shorthand for $X = i in mathbbZ mid 1 le i le n$.
So if $n=0$ then $X=emptyset$.
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
Think of $X = 1,...,n$ as being shorthand for $X = i in mathbbZ mid 1 le i le n$.
So if $n=0$ then $X=emptyset$.
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
answered Jul 16 at 1:23
Lee Mosher
45.7k33478
45.7k33478
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
1
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
add a comment |Â
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
1
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
I understand that but the notation $left 1,...,0right =varnothing $ is what I have a problem with.The set $left 1,...,0right $ is not empty but I'm guesssing it 's not literal.
– user159729
Jul 16 at 1:33
1
1
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
The notation $1,...,n$ is kind of a bad notation, so the way to understand it in the "exceptional" cases is to rewrite it so that it becomes good notation. That's what I was trying to say.
– Lee Mosher
Jul 16 at 1:41
add a comment |Â
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