Toby Inkster > Number-Natural-SetTheory > Number::Natural::SetTheory

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NAME ^

Number::Natural::SetTheory - set-theoretic definition of natural numbers

SYNOPSIS ^

 use Number::Natural::SetTheory qw/:all/;
 my $three = number_to_set(3);
 say (scalar @$three);   # says '3'
 
 # says '0', '1', and '2'
 foreach my $member (@$three)
 {
   say (scalar @$member);
 }
 
 # says '{{},{{}},{{},{{}}}}'
 say set_to_string($three);

DESCRIPTION ^

For years mathematicians struggled to answer what numbers exactly are. A satisfactory answer came out of the world of set theory. Because Perl doesn't have sets as a first class data type, we use arrays instead. The set theory notation for the set of the letters A, B and C is:

  { A, B, C }

The Perlish notation is:

  [ 'A', 'B', 'C' ]

For the rest of this documentation, we'll use Perlish notation unless otherwise stated. Also, it's worth noting that sets are unordered, while arrays are ordered. This module works around that difference by simply ignoring the order of array elements.

Anyway, so what are numbers? We define zero as the empty set:

 our $zero = [];

Further natural numbers are defined as the set containing all smaller natural numbers:

 our $one    = [$zero];
 our $two    = [$zero, $one];
 our $three  = [$zero, $one, $two];
 # etc

This has a nice property:

  scalar @$three == 3

Note that:

  our $not_three = [$zero, $zero, 'Chuck Norris'];
  scalar @$three == 3;   # true

In the case above, the set $not_three does not represent a number at all.

This module offers a number of functions for converting between Perl non-negative integers and the sets representing the natural numbers.

set_is_number($set, $number)

Returns true (see boolean) iff the set represents the number. Also has the property that if $set is an actual Perl scalar integer, it returns true iff the two numbers are equal.

number_to_set($number)

Returns the set that represents a number, given a Perl scalar integer. If $number is not a number, then returns undef.

set_to_number($set)

Converts a set to a Perl scalar integer. Returns undef if the set does not represent a number at all. This is the reverse of $number_to_set.

set_to_string($set)

Returns the set as a string, using number theory notation (curly brackets).

BUGS ^

These functions are very recursive. I wouldn't recommend using them with numbers greater than ten.

This module doesn't really have any use cases.

Please report any bugs to http://rt.cpan.org/Dist/Display.html?Queue=Number-Natural-SetTheory.

SEE ALSO ^

http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers.

AUTHOR ^

Toby Inkster <tobyink@cpan.org>.

COPYRIGHT AND LICENCE ^

This software is copyright (c) 2011-2012 by Toby Inkster.

This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.

DISCLAIMER OF WARRANTIES ^

THIS PACKAGE IS PROVIDED "AS IS" AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.

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