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Defining the number of a class of functions
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Suppose I have a set A and I want to consider all the functions $f:x rightarrow A$ for $x in A$. How do I define the cardinality of the set of such functions?
I can't get my head around the cardinal arithmetic on this; even a hint would be greatly appreciated.
elementary-set-theory
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
asked Jul 17 at 3:36
Mallik
937
add a comment |Â
up vote
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Suppose I have a set A and I want to consider all the functions $f:x rightarrow A$ for $x in A$. How do I define the cardinality of the set of such functions?
I can't get my head around the cardinal arithmetic on this; even a hint would be greatly appreciated.
elementary-set-theory
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
asked Jul 17 at 3:36
Mallik
937
The notation $f:Xto A$ implies $X$ is a set (the domain of $f$), but you seem to imply otherwise?
– gt6989b
Jul 17 at 3:38
@gt6989b I didn't mean to imply otherwise. I can edit it to be lowercase if that would help.
– Mallik
Jul 17 at 3:47
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose I have a set A and I want to consider all the functions $f:x rightarrow A$ for $x in A$. How do I define the cardinality of the set of such functions?
I can't get my head around the cardinal arithmetic on this; even a hint would be greatly appreciated.
elementary-set-theory
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
asked Jul 17 at 3:36
Mallik
937
Suppose I have a set A and I want to consider all the functions $f:x rightarrow A$ for $x in A$. How do I define the cardinality of the set of such functions?
I can't get my head around the cardinal arithmetic on this; even a hint would be greatly appreciated.
elementary-set-theory
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
asked Jul 17 at 3:36
Mallik
937
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
edited Jul 17 at 10:42
Andrés E. Caicedo
63.2k7151236
63.2k7151236
asked Jul 17 at 3:36
Mallik
937
asked Jul 17 at 3:36
Mallik
937
asked Jul 17 at 3:36
Mallik
937
937
The notation $f:Xto A$ implies $X$ is a set (the domain of $f$), but you seem to imply otherwise?
– gt6989b
Jul 17 at 3:38
@gt6989b I didn't mean to imply otherwise. I can edit it to be lowercase if that would help.
– Mallik
Jul 17 at 3:47
add a comment |Â
The notation $f:Xto A$ implies $X$ is a set (the domain of $f$), but you seem to imply otherwise?
– gt6989b
Jul 17 at 3:38
@gt6989b I didn't mean to imply otherwise. I can edit it to be lowercase if that would help.
– Mallik
Jul 17 at 3:47
The notation $f:Xto A$ implies $X$ is a set (the domain of $f$), but you seem to imply otherwise?
– gt6989b
Jul 17 at 3:38
The notation $f:Xto A$ implies $X$ is a set (the domain of $f$), but you seem to imply otherwise?
– gt6989b
Jul 17 at 3:38
@gt6989b I didn't mean to imply otherwise. I can edit it to be lowercase if that would help.
– Mallik
Jul 17 at 3:47
@gt6989b I didn't mean to imply otherwise. I can edit it to be lowercase if that would help.
– Mallik
Jul 17 at 3:47
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
For a given $x in A$, the number of such functions is $|A|^$. If you want the number of such functions for any $x$, simply sum $|A|^$ over $x in A$. Without knowing more about $A$ we cannot simplify further.
answered Jul 17 at 3:58
Bob Krueger
4,0142722
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
For a given $x in A$, the number of such functions is $|A|^$. If you want the number of such functions for any $x$, simply sum $|A|^$ over $x in A$. Without knowing more about $A$ we cannot simplify further.
answered Jul 17 at 3:58
Bob Krueger
4,0142722
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
add a comment |Â
up vote
4
down vote
accepted
For a given $x in A$, the number of such functions is $|A|^$. If you want the number of such functions for any $x$, simply sum $|A|^$ over $x in A$. Without knowing more about $A$ we cannot simplify further.
answered Jul 17 at 3:58
Bob Krueger
4,0142722
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
For a given $x in A$, the number of such functions is $|A|^$. If you want the number of such functions for any $x$, simply sum $|A|^$ over $x in A$. Without knowing more about $A$ we cannot simplify further.
answered Jul 17 at 3:58
Bob Krueger
4,0142722
For a given $x in A$, the number of such functions is $|A|^$. If you want the number of such functions for any $x$, simply sum $|A|^$ over $x in A$. Without knowing more about $A$ we cannot simplify further.
answered Jul 17 at 3:58
Bob Krueger
4,0142722
answered Jul 17 at 3:58
Bob Krueger
4,0142722
answered Jul 17 at 3:58
Bob Krueger
4,0142722
answered Jul 17 at 3:58
Bob Krueger
4,0142722
4,0142722
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
add a comment |Â
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
To sum $|A|^$ over $x in A$, would that be the cardinality of $bigcup_x in A |A|^$?
– Mallik
Jul 17 at 4:42
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
Sure, since for distinct $x$, you necessarily get distinct functions (it’s really a disjoint union).
– Bob Krueger
Jul 17 at 14:26
add a comment |Â
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The notation $f:Xto A$ implies $X$ is a set (the domain of $f$), but you seem to imply otherwise?
– gt6989b
Jul 17 at 3:38
@gt6989b I didn't mean to imply otherwise. I can edit it to be lowercase if that would help.
– Mallik
Jul 17 at 3:47