# man Set::Infinite () - Sets of intervals

## NAME

Set::Infinite - Sets of intervals

## SYNOPSIS

```  use Set::Infinite;
```

```  \$set = Set::Infinite->new(1,2);    # [1..2]
print \$set->union(5,6);            # [1..2],[5..6]
```

## DESCRIPTION

Set::Infinite is a Set Theory module for infinite sets.

A set is a collection of objects. The objects that belong to a set are called its members, or elements.

As objects we allow (almost) anything: reals, integers, and objects (such as dates).

We allow sets to be infinite.

There is no account for the order of elements. For example, {1,2} = {2,1}.

There is no account for repetition of elements. For example, {1,2,2} = {1,1,1,2} = {1,2}.

## new

Creates a new set object:

```    \$set = Set::Infinite->new;             # empty set
\$set = Set::Infinite->new( 10 );       # single element
\$set = Set::Infinite->new( 10, 20 );   # single range
\$set = Set::Infinite->new(
[ 10, 20 ], [ 50, 70 ] );    # two ranges
```
empty set
```    \$set = Set::Infinite->new;
```
set with a single element
```    \$set = Set::Infinite->new( 10 );
```
```    \$set = Set::Infinite->new( [ 10 ] );
```
set with a single span
```    \$set = Set::Infinite->new( 10, 20 );
```
```    \$set = Set::Infinite->new( [ 10, 20 ] );
# 10 <= x <= 20
```
set with a single, open span
```    \$set = Set::Infinite->new(
{
a => 10, open_begin => 0,
b => 20, open_end => 1,
}
);
# 10 <= x < 20
```
set with multiple spans
```    \$set = Set::Infinite->new( 10, 20,  100, 200 );
```
```    \$set = Set::Infinite->new( [ 10, 20 ], [ 100, 200 ] );
```
```    \$set = Set::Infinite->new(
{
a => 10, open_begin => 0,
b => 20, open_end => 0,
},
{
a => 100, open_begin => 0,
b => 200, open_end => 0,
}
);
```

The CWnew() method expects ordered parameters.

If you have unordered ranges, you can build the set using CWunion:

```    @ranges = ( [ 10, 20 ], [ -10, 1 ] );
\$set = Set::Infinite->new;
\$set = \$set->union( @\$_ ) for @ranges;
```

The data structures passed to CWnew must be immutable. So this is not good practice:

```    \$set = Set::Infinite->new( \$object_a, \$object_b );
\$object_a->set_value( 10 );
```

This is the recommended way to do it:

```    \$set = Set::Infinite->new( \$object_a->clone, \$object_b->clone );
\$object_a->set_value( 10 );
```

## clone / copy

Creates a new object, and copy the object data.

## empty_set

Creates an empty set.

If called from an existing set, the empty set inherits the type and density characteristics.

## universal_set

Creates a set containing all possible elements.

If called from an existing set, the universal set inherits the type and density characteristics.

## union

```    \$set = \$set->union(\$b);
```

Returns the set of all elements from both sets.

This function behaves like an OR operation.

```    \$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
\$set2 = new Set::Infinite( [ 7, 20 ] );
print \$set1->union( \$set2 );
# output: [1..4],[7..20]
```

## intersection

```    \$set = \$set->intersection(\$b);
```

Returns the set of elements common to both sets.

This function behaves like an AND operation.

```    \$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
\$set2 = new Set::Infinite( [ 7, 20 ] );
print \$set1->intersection( \$set2 );
# output: [8..12]
```

## difference

```    \$set = \$set->complement;
```

Returns the set of all elements that don't belong to the set.

```    \$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
print \$set1->complement;
# output: (-inf..1),(4..8),(12..inf)
```

The complement function might take a parameter:

```    \$set = \$set->minus(\$b);
```

Returns the set-difference, that is, the elements that don't belong to the given set.

```    \$set1 = new Set::Infinite( [ 1, 4 ], [ 8, 12 ] );
\$set2 = new Set::Infinite( [ 7, 20 ] );
print \$set1->minus( \$set2 );
# output: [1..4]
```

## simmetric_difference

Returns a set containing elements that are in either set, but not in both. This is the set version of XOR.

## real

```    \$set1 = \$set->real;
```

Returns a set with density 0.

## integer

```    \$set1 = \$set->integer;
```

Returns a set with density 1.

## intersects

```    \$logic = \$set->intersects(\$b);
```

## contains

```    \$logic = \$set->contains(\$b);
```

## is_null

```    \$logic = \$set->is_null;
```

## is_nonempty

This set that has at least 1 element.

## is_span

This set that has a single span or interval.

## is_singleton

This set that has a single element. Every element of this set is a member of the given set. Every element of this set is a member of the given set. Some members of the given set are not elements of this set. The given set has no elements in common with this set.

## is_too_complex

Sometimes a set might be too complex to enumerate or print.

This happens with sets that represent infinite recurrences, such as when you ask for a quantization on a set bounded by -inf or inf.

## min

```    \$i = \$set->min;
```

## max

```    \$i = \$set->max;
```

## size

```    \$i = \$set->size;
```

## count

```    \$i = \$set->count;
```

## stringification

```    print \$set;
```

```    \$str = "\$set";
```

## comparison

```    sort
```

```    > < == >= <= <=>
```

## CLASS METHODS

```    Set::Infinite->separators(@i)
```

```        chooses the interval separators for stringification.
```

```        default are [ ] ( ) '..' ','.
```

```    inf
```

```        returns an 'Infinity' number.
```

```    minus_inf
```

```        returns '-Infinity' number.
```

## type

```    type( "My::Class::Name" )
```

Chooses a default object data type.

Default is none (a normal Perl SCALAR).

## span

```    \$set1 = \$set->span;
```

Returns the set span.

## until

Extends a set until another:

```    0,5,7 -> until 2,6,10
```

gives

```    [0..2), [5..6), [7..10)
```

## end_set

These methods do the inverse of the until method.

Given:

```    [0..2), [5..6), [7..10)
```

start_set is:

```    0,5,7
```

end_set is:

```    2,6,10
```

## intersected_spans

```    \$set = \$set1->intersected_spans( \$set2 );
```

The method returns a new set, containing all spans that are intersected by the given set.

Unlike the CWintersection method, the spans are not modified. See diagram below:

```               set1   [....]   [....]   [....]   [....]
set2      [................]
```

```       intersection      [.]   [....]   [.]
```

```  intersected_spans   [....]   [....]   [....]
```

## quantize

```    quantize( parameters )
```

```        Makes equal-sized subsets.
```

```        Returns an ordered set of equal-sized subsets.
```

```        Example:
```

```            \$set = Set::Infinite->new([1,3]);
print join (" ", \$set->quantize( quant => 1 ) );
```

```        Gives:
```

```            [1..2) [2..3) [3..4)
```

## select

```    select( parameters )
```

Selects set spans based on their ordered positions

CWselect has a behaviour similar to an array CWslice.

```            by       - default=All
count    - default=Infinity
```

``` 0  1  2  3  4  5  6  7  8      # original set
0  1  2                        # count => 3
1              6            # by => [ -2, 1 ]
```

## offset

```    offset ( parameters )
```

```        Offsets the subsets. Parameters:
```

```            value   - default=[0,0]
mode    - default='offset'. Possible values are: 'offset', 'begin', 'end'.
unit    - type of value. Can be 'days', 'weeks', 'hours', 'minutes', 'seconds'.
```

## iterate

```    iterate ( sub { } , @args )
```

Iterates on the set spans, over a callback subroutine. Returns the union of all partial results.

The callback argument CW\$_[0] is a span. If there are additional arguments they are passed to the callback.

The callback can return a span, a hashref (see CWSet::Infinite::Basic), a scalar, an object, or CWundef.

[EXPERIMENTAL] CWiterate accepts a CWbacktrack_callback argument. This can be used when the data needs to be processed in some special way while backtracking. The syntax is:

```    iterate ( sub { } , backtrack_callback => sub { }, @args )
```

## first / last

```    first / last
```

In scalar context returns the first or last interval of a set.

In list context returns the first or last interval of a set, and the remaining set (the 'tail').

## type

```    type( "My::Class::Name" )
```

Chooses a default object data type.

default is none (a normal perl SCALAR).

## _cleanup

```    \$set->_cleanup;
```

Internal function to fix the internal set representation. This is used after operations that might return invalid values.

## _backtrack

```    \$set->_backtrack( 'intersection', \$b );
```

Internal function to evaluate recurrences.

## numeric

```    \$set->numeric;
```

Internal function to ignore the set type. It is used in some internal optimizations, when it is possible to use scalar values instead of objects.

## fixtype

```    \$set->fixtype;
```

Internal function to fix the result of operations that use the numeric() function.

## tolerance

```    \$set = \$set->tolerance(0)    # defaults to real sets (default)
\$set = \$set->tolerance(1)    # defaults to integer sets
```

Internal function for changing the set density.

## min_a

```    (\$min, \$min_is_open) = \$set->min_a;
```

## max_a

```    (\$max, \$max_is_open) = \$set->max_a;
```

## as_string

Implements the stringification operator.

Stringification of unbounded recurrences is not implemented.

Unbounded recurrences are stringified as function descriptions, if the class variable CW\$PRETTY_PRINT is set.

## spaceship

Implements the comparison operator.

Comparison of unbounded recurrences is not implemented.

## CAVEATS

```    \$set = Set::Infinite->new(10,1);
```

Will be interpreted as [1..10]

```    \$set = Set::Infinite->new(1,2,3,4);
```
Will be interpreted as [1..2],[3..4] instead of [1,2,3,4]. You probably want ->new([1],[2],[3],[4]) instead, or maybe ->new(1,4)
```    \$set = Set::Infinite->new(1..3);
```
Will be interpreted as [1..2],3 instead of [1,2,3]. You probably want ->new(1,3) instead.

## INTERNALS

The base set object, without recurrences, is a CWSet::Infinite::Basic.

A recurrence-set is represented by a method name, one or two parent objects, and extra arguments. The CWlist key is set to an empty array, and the CWtoo_complex key is set to CW1.

This is a structure that holds the union of two complex sets:

```  {
too_complex => 1,             # "this is a recurrence"
list   => [ ],                # not used
method => 'union',            # function name
parent => [ \$set1, \$set2 ],   # "leaves" in the syntax-tree
param  => [ ]                 # optional arguments for the function
}
```

This is a structure that holds the complement of a complex set:

```  {
too_complex => 1,             # "this is a recurrence"
list   => [ ],                # not used
method => 'complement',       # function name
parent => \$set,               # "leaf" in the syntax-tree
param  => [ ]                 # optional arguments for the function
}
```

See modules DateTime::Set, DateTime::Event::Recurrence, DateTime::Event::ICal, DateTime::Event::Cron for up-to-date information on date-sets.

The perl-date-time project <http://datetime.perl.org>

## AUTHOR

Flavio Soibelmann Glock <fglock@pucrs.br>