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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. However it differs from Intervals in which it's aim is not on numerical precision and calculation, but on ease of use and solving some general interval related problems.

Install

You may install it traditionally, tipically 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 extension.

  nil.to_interval     # -> Interval[]
  1.to_interval       # -> Interval[1]
  (1..2).to_interval  # -> Interval[1,2]
  (1...3).to_interval # -> Interval[1,2]
  [1,2].to_interval   # -> Interval[1,2]

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

Operations

  • Complement

#complement, alias: #~

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

#union, aliases: #|,#+

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

#intersect, aliases: #&,#^

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

#minus, alias: #-

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

Generic Intervals

You may use any Comparable class:

  • String
Interval['a','c'] ^ Interval['b','d'] # -> Interval['b','c']
Interval['a','c'] | Interval['b','d'] # -> Interval['a','d']
  • Time
def t(seconds)
  Time.now + seconds
end

Interval[t(1),t(5)] ^ Interval[t(3),t(7)] # -> Interval[t(3),t(5)]
Interval[t(1),t(2)] | Interval[t(3),t(4)] # -> Interval[[t(1),t(2)],[t(3),t(4)]]

Math Extensions

By requiring borel/math_extensions you are provided with some natural math-related interval methods:

require 'borel/math_extensions'

Interval[1,5].rand  # -> Random.new.rand 1..5
Interval[1,5].width # -> 5-1, only for simple intervals

It's supported only for Numeric, Comparable and arithmetic supported classes

Remarks

  • There is no distinction between open and closed intervals
  • complement and minus operations, and also Math Extensions have limited support for non numeric-comparable classes

License

(The MIT License)

Copyright (c) 2012 Amadeus Folego

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

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