如何证明所有两个参数函数的集合都是不可数的

Question

We can prove that set of all one argument functions cannot be countable using the cantor's diagonal. for example

     1    2    3    4    5    6    7 ......

f1   10   12   23   1    3    12   3 ......    
f2   15    6    7   8    9    11   4 ...... 
f3   14    2    4   3    3     4   5 ...... 
f4   12    2    3   5    1    20   56 .....   
.
.
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for all functions f1 to fn we can pass all the arguments and 1 to n for some n. then by taking the diagonal values and add 1 to diagonal values and we can prove that we can't count all the one argument functions.(since change the diagonal values will produce a row unique which haven't listed)

Wonder is there a particular method to count two argument functions also??..

Thanks..

Answer 1

Wonder is there a particular method to count two argument functions also??..

You mean Wonder is there a particular method to count two argument functions? ("also" would imply that one exists for one-argument functions).

If a subset of a non-countable set is always also noncountable then you could just use the subset of the set of all two-argument functions where you fix the second parameter to a constant (making it essentially equal to a one-argument-function). However I doubt that this assumption is true.

I think you left out some important prerequisites about the diagram (how the fn are constructed as they are not chosen arbitrarily). Maybe examining them will lead you to the clue? I guess this is a homework question? Is it allowed on stackoverflow to post homework questions and in your university to have them let solved by someone else?

Answer 2

I think I've found an answer. I'm writing the answer in case anyone is interested.

we can prove than all the pairs of positive integers can be countable. see below

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6).....
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6).....
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6).....
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6).....
 .
 .
 .

so from the cantor's zig zag we can prove that they can be countable.. see page 8 on this book http://www.scribd.com/doc/51068193/3/Enumerable-Sets

(1,1) (1,2) (2,1) (1,3) (2,2) (3,1) ....

so we can write our problem as below.

    (1,1)   (1,2)   (2,1)   (1,3)   (2,2)   (3,1)

f1   10      12       23       1      3       12    ......    
f2   15       6        7       8      9       11    ...... 
f3   14       2        4       3      3        4    ...... 
f4   12       2        3       5      1       20    ......   
.
.
.

Now by the knowledge of cantor's diagonal.. we can argue that all the two argument functions can't be countable.