Borel

Borelian sets are formed by enumerable union, intersection or complement, of intervals.

Borel enables performing regular operations on intervals of any comparable class.

Borel borrows many of the ideas (and code) from the Intervals gem.

Installation

You may install it traditionally, for interactive sessions:

gem install borel

Or just put this somewhere on your application's Gemfile

gem 'borel'

Usage

Initializing

An Interval can be initialized with an empty, one or two sized array (respectively for an empty, degenerate or simple interval), or an array of one or two sized arrays (for a multiple interval).

  Interval[]
  Interval[1]
  Interval[0,1]
  Interval[[0,1],[2,3],[5]]

Another way to initialize an Interval is by using the to_interval method on Ranges or Numbers.

  1.to_interval
  (0..1).to_interval
  (0...2).to_interval

The Infinity constant is available for specifying intervals with no upper or lower boundary.

  Interval[-Infinity, 0]
  Interval[1, Infinity]
  Interval[-Infinity, Infinity]

Properties

Some natural properties of intervals:

  Interval[1].degenerate?       # -> true
  Interval[[0,1],[2,3]].simple? # -> false
  Interval[].empty?             # -> true
  Interval[1,5].include?(3.4)   # -> true

Operations

• Complement

complement and ~

    ~Interval[0,5]             # -> Interval[[-Infinity, 0], [5, Infinity]]

• Union

union, | and +

Interval[0,5] | Interval[-1,3] # -> Interval[-1,5]

• Intersection

intersect, &, ^

Interval[0,5] ^ Interval[-1,3] # -> Interval[0,3]

• Subtraction

minus and -

Interval[0,5] - Interval[-1,3] # -> Interval[3,5]

Classes of Intervals

You may use any comparable class

Interval['a','c'] ^ Interval['b','d'] # -> Interval['b','c']
Interval['a','c'] | Interval['b','d'] # -> Interval['a','d']

Remarks

• There is no distinction between open and closed intervals
• Complement and Minus operations are not supported on any class