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Welcome to the Wikipedia Mathematics Reference Desk Archives |
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Hello,
I'm dealing with what I believe is an intractable problem. Here it is stated as succinctly as possible: A positive integer in its letter form can equal itself (given that a is 1, b is 2, and so on) when the sum of the letters are taken. What is the second one? Naturally, I want to be exhaustive in the answers I can give: does one write the number in traditional English as "TWENTY FIVE", "ONE HUNDRED AND THIRTY SIX", does one leave out "and", or does one resort to calling out each number within the greater string of numbers, e.g., "111" would be "ONE ONE ONE"? Besides that linguistic dimension of the problem, there is also the immense difficulty of finding the damn numbers! I'm trying to find these answers quickly and efficiently, but seeing how open-ended the problem is (but don't think there isn't an answer, because there definitely is), I'm wondering if anyone can provide a suitable approach to tackle this problem, even if I have to learn to write a program to do it. Thanks. I know this isn't your everyday sort of problem, and it probably will require a lot of thought. Another possibility that I don't know about is whether such a number would have a certain, known property, like an "honest number" or perfect number. If anyone knows of such numbers without having to go through the bricks and mortar of finding the answer, then please let me know that as well. Coitoergosum ( talk) 02:02, 26 April 2010 (UTC)
Hi all, My problem is little but very frustrating. I am reading from a book that has very little explanations and is not very rigorous too, and am stuck at a little point. I belive there is a misprint but then I can't make out what to substitute for the correct values: Consider the equation (in Z) (k,b are even). The book says that is a solution, which I fail to see. I believe the -1 at the end of the xk term is an error. But reading onward it is evident that the -1 is very important in the subsequent discussion and in fact the value of xk must be exactly what is proposed. So I feel the error must be in defining the other xi terms. Subject to the additional contraints that xi's don't belong to [at+1,at+1], at's are increasing and how should I modify the solution. i.e. the xi's. Also is some modification actually needed or I am just plain missing something. Thanks- Shahab ( talk) 05:35, 26 April 2010 (UTC)
Hello. I'm looking for information on the best way to factorise . I notice that swapping a and b negates the result, which suggests that the expression is divisible by (a-b), but I'm not really sure how to proceed from there. The people on IRC advised that considering a, b, and c as points in three-dimensional space and employing Lagrange multipliers could work, but I don't know much about that. Thanks for the help. — Anonymous Dissident Talk 12:18, 26 April 2010 (UTC)
The article says that "irrational algebraic numbers and functions are themselves expressible as integrals of rational functions over rational domains". Could someone handwave how algebraic numbers can be expressed that way? —Preceding unsigned comment added by 212.87.13.69 ( talk) 16:54, 26 April 2010 (UTC)
The game of cricket has a team, usually 11 in number, who in an innings bat in pairs, initially person 1 and person 2 together. Whoever is dismissed in a partnership is replaced by the next person in numerical order, this continuing until the 10th partnership is broken, when the innings is complete. If there were only 3 players there would be 2 possible partnership sequences, (1,2), (1,3) or (1,2), (2,3). A 4th player would double the possible number of sequences to these: (1,2), (1,3), (1,4) or (1,2), (1,3), (3,4) or (1,2), (2,3), (2,4) or (1,2), (2,3), (3,4). It's apparent that for n players there will be 2^(n-2) different sequences of length n-1. My problem - to get an algorithm to generate them. I feel that this should be fairly standard, but can't get one to work.→ 86.166.205.252 ( talk) 20:32, 26 April 2010 (UTC)
program CRICKET
character*16 BIN
integer*2 I,J,A,B,N
print *,"Enter N (2-16):"
read (*,*) N
do I = 1, 2**(N-2)
A = 1
B = 2
print *,"(",A,",",B,")"
J = I-1
call BINARY(J, BIN) ! Returns 16 character binary string, 5 = "0000000000000101"
do J = 19-N,16
if (BIN(J:J) .eq. "0") B = B+1
if (BIN(J:J) .eq. "1") then
A = B
B = A+1
endif
print *,"(",A,",",B,")"
enddo
print *," "
enddo
end
program CRICKET2
integer*2 N,A,B,I
character*178 STRING
print *,"Enter N (>2):"
read (*,*) N
I = 2
A = 1
B = 2
write (STRING,*) "(",A,",",B,")"
if (N .ge. 2) call LOOP (N, A,B,I,STRING )
end
******************************************
subroutine LOOP (N, A,B,I,STRING )
integer*2 N,A,B,I
character*178 STRING
integer*2 OLD_B,OLD_I
character*178 OLD_STRING
if (I .lt. N) then
OLD_B = B
OLD_I = I
OLD_STRING = STRING
I = OLD_I+1
B = OLD_B+1
write (STRING(LEN_TRIM(STRING)+2:),*) "(",A,",",B,")"
call LOOP (N, A,B,I,STRING )
I = OLD_I+1
A = OLD_B
B = A+1
STRING = OLD_STRING
write (STRING(LEN_TRIM(STRING)+2:),*) "(",A,",",B,")"
call LOOP (N, A,B,I,STRING )
else
print *,STRING(:LEN_TRIM(STRING))
endif
return
end
Enter N (>2): 2 ( 1, 2)
Enter N (>2): 3 ( 1, 2) ( 1, 3) ( 1, 2) ( 2, 3)
Enter N (>2): 4 ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 2) ( 1, 3) ( 3, 4) ( 1, 2) ( 2, 3) ( 2, 4) ( 1, 2) ( 2, 3) ( 3, 4)
Enter N (>2): 5 ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 5) ( 1, 2) ( 1, 3) ( 1, 4) ( 4, 5) ( 1, 2) ( 1, 3) ( 3, 4) ( 3, 5) ( 1, 2) ( 1, 3) ( 3, 4) ( 4, 5) ( 1, 2) ( 2, 3) ( 2, 4) ( 2, 5) ( 1, 2) ( 2, 3) ( 2, 4) ( 4, 5) ( 1, 2) ( 2, 3) ( 3, 4) ( 3, 5) ( 1, 2) ( 2, 3) ( 3, 4) ( 4, 5)
Enter N (>2): 6 ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 5) ( 1, 6) ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 5) ( 5, 6) ( 1, 2) ( 1, 3) ( 1, 4) ( 4, 5) ( 4, 6) ( 1, 2) ( 1, 3) ( 1, 4) ( 4, 5) ( 5, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 3, 5) ( 3, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 3, 5) ( 5, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 4, 5) ( 4, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 4, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 2, 5) ( 2, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 2, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 4, 5) ( 4, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 4, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 3, 4) ( 3, 5) ( 3, 6) ( 1, 2) ( 2, 3) ( 3, 4) ( 3, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 3, 4) ( 4, 5) ( 4, 6)
template<int P1, int P2, typename Prev=void>
struct Pair : Prev { enum { first = P1, second = P2 }; };
template<> struct Pair<1,2> { enum { first = 1, second = 2 }; };
template<int Max, typename P = Pair<1,2>, int Next = (P::first > P::second ? P::first : P::second)>
struct Seq : Seq<Max, Pair<P::first, Next+1 , P> >, Seq<Max, Pair<Next+1, P::second, P> > {};
template<int Max,typename P> struct Seq<Max, P, Max> { typedef P result; };
int main()
{
Seq<5>::result();
}
/* Output: sourceFile.cpp(13) : error C2385: ambiguous access of 'sequence'
could be the 'sequence' in base 'Seq<5,Pair<1,5,Pair<1,4,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,4,Pair<1,4,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<4,5,Pair<4,3,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,3,Pair<4,3,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<3,5,Pair<3,4,Pair<3,2,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,4,Pair<3,4,Pair<3,2,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<4,5,Pair<4,2,Pair<3,2,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,2,Pair<4,2,Pair<3,2,Pair<1,2,void> > > >,5>'
*/
Ouch, decltype, that was brilliant but cruel ;-). I like Haskell for problems like this:
cricket a,b = [[(a,b)]]
cricket (a:b:c:ds) = [(a,b):es | es<-cricket (b:c:ds) ++ cricket (a:c:ds)]
main = mapM_ print $ cricket 1..4
{- output:
[(1,2),(2,3),(3,4)]
[(1,2),(2,3),(2,4)]
[(1,2),(1,3),(3,4)]
[(1,2),(1,3),(1,4)]
-}
This is recursive: the "cricket [a,b]" equation handles the 2-element case, and "cricket (a:b:c:ds)" handles the recursive case (3 elements a,b,c followed by a possibly empty remainder of the list). The bracketed expression in the recursive case is a list comprehensions. The mapM_ in the "main" function runs "print" on each of the sublists, so they are shown one per line. [1..4] is shorthand for [1,2,3,4]. 69.228.170.24 ( talk) 09:07, 2 May 2010 (UTC)
hi the problem i have says "50 people build 10 bridges in 500 hours. If 15 people build 40 bridges, how many hours will they take to make them?" how do i solve this?? —Preceding unsigned comment added by Needlotsofhelp ( talk • contribs) 21:38, 26 April 2010 (UTC)
well my teacher told me about a way to find the answer using a fomrula but i dont remember it was it like multiply people by hours and then divide by stuff they built??? i'm not sure.... —Preceding unsigned comment added by Needlotsofhelp ( talk • contribs) 22:20, 26 April 2010 (UTC)
Well? -- 128.62.32.123 ( talk) 23:00, 26 April 2010 (UTC)
Is it possible to solve...
((ay^2/(b+c))-cy)*(1-(y^2/b^2))^0.5 dy from y=b to -b
Many thanks for any help. Here is a link to it on wolfram alpha if that helps to visualise the problem [1].
From Andrew McArthur -- 137.222.114.238 ( talk) 23:00, 26 April 2010 (UTC)
If I'm reading the question right, the problem is to evaluate
I'd separate it thus:
The second integral is that of an odd function over an interval symmetric about the origin; therefore the second integral is zero. The first integral is that of an even function over an interval symmetric about the origin; therefore its value is twice that of the integral over the half-interval:
Now let
etc. (And of course θ will go from 0 to π/2.) Michael Hardy ( talk) 17:28, 27 April 2010 (UTC)
The standard definition for two sets X, Y of | X | ≤ | Y | is that there is an injection from X to Y. An alternative definition may be that there is a surjection from Y to X; under the assumption of the Axiom of Choice, this latter definition is equivalent. How does this definition behave if the axiom of choice isn't assumed? -- 128.62.32.123 ( talk) 23:26, 26 April 2010 (UTC)
Ask user:Trovatore about this one. Michael Hardy ( talk) 17:35, 27 April 2010 (UTC)
I notice that mathematicians seem to have difficulty with the notion of an open-ended question. So far, only one point has been brought up that I didn't specifically mention in trying to give some examples of what I'm looking for. -- 128.62.55.11 ( talk) 00:24, 28 April 2010 (UTC)
Thanks, people who made an attempt, I guess. I was hoping for more. I'd be very surprised if no one has ever studied this alternative definition in far more depth than the responses here seem to indicate. I'll assume the lack of further answers and insights indicate that no one else has anything substantial to add. I'll see if I can hunt down some real mathematicians who know more about this, since no one here seems to. -- 128.62.45.2 ( talk) 18:00, 28 April 2010 (UTC)
Mathematics desk | ||
---|---|---|
< April 25 | << Mar | April | May >> | April 27 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Hello,
I'm dealing with what I believe is an intractable problem. Here it is stated as succinctly as possible: A positive integer in its letter form can equal itself (given that a is 1, b is 2, and so on) when the sum of the letters are taken. What is the second one? Naturally, I want to be exhaustive in the answers I can give: does one write the number in traditional English as "TWENTY FIVE", "ONE HUNDRED AND THIRTY SIX", does one leave out "and", or does one resort to calling out each number within the greater string of numbers, e.g., "111" would be "ONE ONE ONE"? Besides that linguistic dimension of the problem, there is also the immense difficulty of finding the damn numbers! I'm trying to find these answers quickly and efficiently, but seeing how open-ended the problem is (but don't think there isn't an answer, because there definitely is), I'm wondering if anyone can provide a suitable approach to tackle this problem, even if I have to learn to write a program to do it. Thanks. I know this isn't your everyday sort of problem, and it probably will require a lot of thought. Another possibility that I don't know about is whether such a number would have a certain, known property, like an "honest number" or perfect number. If anyone knows of such numbers without having to go through the bricks and mortar of finding the answer, then please let me know that as well. Coitoergosum ( talk) 02:02, 26 April 2010 (UTC)
Hi all, My problem is little but very frustrating. I am reading from a book that has very little explanations and is not very rigorous too, and am stuck at a little point. I belive there is a misprint but then I can't make out what to substitute for the correct values: Consider the equation (in Z) (k,b are even). The book says that is a solution, which I fail to see. I believe the -1 at the end of the xk term is an error. But reading onward it is evident that the -1 is very important in the subsequent discussion and in fact the value of xk must be exactly what is proposed. So I feel the error must be in defining the other xi terms. Subject to the additional contraints that xi's don't belong to [at+1,at+1], at's are increasing and how should I modify the solution. i.e. the xi's. Also is some modification actually needed or I am just plain missing something. Thanks- Shahab ( talk) 05:35, 26 April 2010 (UTC)
Hello. I'm looking for information on the best way to factorise . I notice that swapping a and b negates the result, which suggests that the expression is divisible by (a-b), but I'm not really sure how to proceed from there. The people on IRC advised that considering a, b, and c as points in three-dimensional space and employing Lagrange multipliers could work, but I don't know much about that. Thanks for the help. — Anonymous Dissident Talk 12:18, 26 April 2010 (UTC)
The article says that "irrational algebraic numbers and functions are themselves expressible as integrals of rational functions over rational domains". Could someone handwave how algebraic numbers can be expressed that way? —Preceding unsigned comment added by 212.87.13.69 ( talk) 16:54, 26 April 2010 (UTC)
The game of cricket has a team, usually 11 in number, who in an innings bat in pairs, initially person 1 and person 2 together. Whoever is dismissed in a partnership is replaced by the next person in numerical order, this continuing until the 10th partnership is broken, when the innings is complete. If there were only 3 players there would be 2 possible partnership sequences, (1,2), (1,3) or (1,2), (2,3). A 4th player would double the possible number of sequences to these: (1,2), (1,3), (1,4) or (1,2), (1,3), (3,4) or (1,2), (2,3), (2,4) or (1,2), (2,3), (3,4). It's apparent that for n players there will be 2^(n-2) different sequences of length n-1. My problem - to get an algorithm to generate them. I feel that this should be fairly standard, but can't get one to work.→ 86.166.205.252 ( talk) 20:32, 26 April 2010 (UTC)
program CRICKET
character*16 BIN
integer*2 I,J,A,B,N
print *,"Enter N (2-16):"
read (*,*) N
do I = 1, 2**(N-2)
A = 1
B = 2
print *,"(",A,",",B,")"
J = I-1
call BINARY(J, BIN) ! Returns 16 character binary string, 5 = "0000000000000101"
do J = 19-N,16
if (BIN(J:J) .eq. "0") B = B+1
if (BIN(J:J) .eq. "1") then
A = B
B = A+1
endif
print *,"(",A,",",B,")"
enddo
print *," "
enddo
end
program CRICKET2
integer*2 N,A,B,I
character*178 STRING
print *,"Enter N (>2):"
read (*,*) N
I = 2
A = 1
B = 2
write (STRING,*) "(",A,",",B,")"
if (N .ge. 2) call LOOP (N, A,B,I,STRING )
end
******************************************
subroutine LOOP (N, A,B,I,STRING )
integer*2 N,A,B,I
character*178 STRING
integer*2 OLD_B,OLD_I
character*178 OLD_STRING
if (I .lt. N) then
OLD_B = B
OLD_I = I
OLD_STRING = STRING
I = OLD_I+1
B = OLD_B+1
write (STRING(LEN_TRIM(STRING)+2:),*) "(",A,",",B,")"
call LOOP (N, A,B,I,STRING )
I = OLD_I+1
A = OLD_B
B = A+1
STRING = OLD_STRING
write (STRING(LEN_TRIM(STRING)+2:),*) "(",A,",",B,")"
call LOOP (N, A,B,I,STRING )
else
print *,STRING(:LEN_TRIM(STRING))
endif
return
end
Enter N (>2): 2 ( 1, 2)
Enter N (>2): 3 ( 1, 2) ( 1, 3) ( 1, 2) ( 2, 3)
Enter N (>2): 4 ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 2) ( 1, 3) ( 3, 4) ( 1, 2) ( 2, 3) ( 2, 4) ( 1, 2) ( 2, 3) ( 3, 4)
Enter N (>2): 5 ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 5) ( 1, 2) ( 1, 3) ( 1, 4) ( 4, 5) ( 1, 2) ( 1, 3) ( 3, 4) ( 3, 5) ( 1, 2) ( 1, 3) ( 3, 4) ( 4, 5) ( 1, 2) ( 2, 3) ( 2, 4) ( 2, 5) ( 1, 2) ( 2, 3) ( 2, 4) ( 4, 5) ( 1, 2) ( 2, 3) ( 3, 4) ( 3, 5) ( 1, 2) ( 2, 3) ( 3, 4) ( 4, 5)
Enter N (>2): 6 ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 5) ( 1, 6) ( 1, 2) ( 1, 3) ( 1, 4) ( 1, 5) ( 5, 6) ( 1, 2) ( 1, 3) ( 1, 4) ( 4, 5) ( 4, 6) ( 1, 2) ( 1, 3) ( 1, 4) ( 4, 5) ( 5, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 3, 5) ( 3, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 3, 5) ( 5, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 4, 5) ( 4, 6) ( 1, 2) ( 1, 3) ( 3, 4) ( 4, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 2, 5) ( 2, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 2, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 4, 5) ( 4, 6) ( 1, 2) ( 2, 3) ( 2, 4) ( 4, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 3, 4) ( 3, 5) ( 3, 6) ( 1, 2) ( 2, 3) ( 3, 4) ( 3, 5) ( 5, 6) ( 1, 2) ( 2, 3) ( 3, 4) ( 4, 5) ( 4, 6)
template<int P1, int P2, typename Prev=void>
struct Pair : Prev { enum { first = P1, second = P2 }; };
template<> struct Pair<1,2> { enum { first = 1, second = 2 }; };
template<int Max, typename P = Pair<1,2>, int Next = (P::first > P::second ? P::first : P::second)>
struct Seq : Seq<Max, Pair<P::first, Next+1 , P> >, Seq<Max, Pair<Next+1, P::second, P> > {};
template<int Max,typename P> struct Seq<Max, P, Max> { typedef P result; };
int main()
{
Seq<5>::result();
}
/* Output: sourceFile.cpp(13) : error C2385: ambiguous access of 'sequence'
could be the 'sequence' in base 'Seq<5,Pair<1,5,Pair<1,4,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,4,Pair<1,4,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<4,5,Pair<4,3,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,3,Pair<4,3,Pair<1,3,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<3,5,Pair<3,4,Pair<3,2,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,4,Pair<3,4,Pair<3,2,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<4,5,Pair<4,2,Pair<3,2,Pair<1,2,void> > > >,5>'
or could be the 'sequence' in base 'Seq<5,Pair<5,2,Pair<4,2,Pair<3,2,Pair<1,2,void> > > >,5>'
*/
Ouch, decltype, that was brilliant but cruel ;-). I like Haskell for problems like this:
cricket a,b = [[(a,b)]]
cricket (a:b:c:ds) = [(a,b):es | es<-cricket (b:c:ds) ++ cricket (a:c:ds)]
main = mapM_ print $ cricket 1..4
{- output:
[(1,2),(2,3),(3,4)]
[(1,2),(2,3),(2,4)]
[(1,2),(1,3),(3,4)]
[(1,2),(1,3),(1,4)]
-}
This is recursive: the "cricket [a,b]" equation handles the 2-element case, and "cricket (a:b:c:ds)" handles the recursive case (3 elements a,b,c followed by a possibly empty remainder of the list). The bracketed expression in the recursive case is a list comprehensions. The mapM_ in the "main" function runs "print" on each of the sublists, so they are shown one per line. [1..4] is shorthand for [1,2,3,4]. 69.228.170.24 ( talk) 09:07, 2 May 2010 (UTC)
hi the problem i have says "50 people build 10 bridges in 500 hours. If 15 people build 40 bridges, how many hours will they take to make them?" how do i solve this?? —Preceding unsigned comment added by Needlotsofhelp ( talk • contribs) 21:38, 26 April 2010 (UTC)
well my teacher told me about a way to find the answer using a fomrula but i dont remember it was it like multiply people by hours and then divide by stuff they built??? i'm not sure.... —Preceding unsigned comment added by Needlotsofhelp ( talk • contribs) 22:20, 26 April 2010 (UTC)
Well? -- 128.62.32.123 ( talk) 23:00, 26 April 2010 (UTC)
Is it possible to solve...
((ay^2/(b+c))-cy)*(1-(y^2/b^2))^0.5 dy from y=b to -b
Many thanks for any help. Here is a link to it on wolfram alpha if that helps to visualise the problem [1].
From Andrew McArthur -- 137.222.114.238 ( talk) 23:00, 26 April 2010 (UTC)
If I'm reading the question right, the problem is to evaluate
I'd separate it thus:
The second integral is that of an odd function over an interval symmetric about the origin; therefore the second integral is zero. The first integral is that of an even function over an interval symmetric about the origin; therefore its value is twice that of the integral over the half-interval:
Now let
etc. (And of course θ will go from 0 to π/2.) Michael Hardy ( talk) 17:28, 27 April 2010 (UTC)
The standard definition for two sets X, Y of | X | ≤ | Y | is that there is an injection from X to Y. An alternative definition may be that there is a surjection from Y to X; under the assumption of the Axiom of Choice, this latter definition is equivalent. How does this definition behave if the axiom of choice isn't assumed? -- 128.62.32.123 ( talk) 23:26, 26 April 2010 (UTC)
Ask user:Trovatore about this one. Michael Hardy ( talk) 17:35, 27 April 2010 (UTC)
I notice that mathematicians seem to have difficulty with the notion of an open-ended question. So far, only one point has been brought up that I didn't specifically mention in trying to give some examples of what I'm looking for. -- 128.62.55.11 ( talk) 00:24, 28 April 2010 (UTC)
Thanks, people who made an attempt, I guess. I was hoping for more. I'd be very surprised if no one has ever studied this alternative definition in far more depth than the responses here seem to indicate. I'll assume the lack of further answers and insights indicate that no one else has anything substantial to add. I'll see if I can hunt down some real mathematicians who know more about this, since no one here seems to. -- 128.62.45.2 ( talk) 18:00, 28 April 2010 (UTC)