[index] [finite-map] Algebra::Set / Enumerable

Algebra::Set

Class of Set

This is the class of sets. The conclusion relationship is determined by each and has?, that is, s is the subset of t if and only if

s.all?{|x| t.has?(x)}

is true.

File Name:

• finite-set.rb

SuperClass:

• Object

Included Module:

• Enumerable

Class Methods:

::[[obj0, [obj1, [obj2, ...]]]]

Creates Set objects from parameters.

Example: Create {"a", [1, 2], 0}

require "finite-set"
p Algebra::Set[0, "a", [1, 2]]
p Algebra::Set.new(0, "a", [1, 2])
p Algebra::Set.new_a([0, "a", [1, 2]])
p Algebra::Set.new_h({0=>true, "a"=>true, [1, 2]=>true})

::new([obj0, [obj1, [obj2, ...]]])

Creates Set objects from parameters.

::new_a(a)

Creates Set objects from an array a.

::new_h(h)

Creates Set objects from a hash.

::empty_set

Returns the empty set.

::phi

::null

Alias of ::empty_set.

::singleton(x)

Creates the set of one element x.

Methods:

empty_set

Returns the empty set.

phi

null

Alias of ::empty_set.

empty?

Returns true if self is the empty set.

phi?

empty_set?

null?

Alias of empty?

singleton(x)

Creates the set of one element x.

singleton?

Returns true if self is a singleton set.

size

Returns the size of self.

each

Iterates the block with the block parameter of each element. The order of iteration is indefinite.

Example:

require "finite-set"
include Algebra
Set[0, 1, 2].each do |x|
  p x #=> 1, 0, 2
end

separate

Return the set of the elements which make the block true.

Example:

require "finite-set"
include Algebra
p Set[0, 1, 2, 3].separate{|x| x % 2 == 0} #=> {2, 0}

select_s

find_all_s

Alias of separate

map_s

Return the set of the values of the block.

Example:

require "finite-set"
include Algebra
p Set[0, 1, 2, 3].map_s{|x| x % 2 + 1} #=> {2, 1}

pick

Returns a elements of self. The chois is indefinite.

shift

Takes an element from self and returns it.

Example:

require "finite-set"
include Algebra
s = Set[0, 1, 2, 3]
p s.shift #=> 2
p s #=> {0, 1, 3}

dup

Returns the duplication of self.

append!(x)

Appends x to self and returns self.

push

<<

Alias of append!.

append(x)

Duplicates and appends x to it and returns it.

concat(other)

Add the all elements of other. This is the destructive version of +.

rehash

Rehashes the internal Hash object.

eql?(other)

Returns true if self is equal to other. This is equivalent to self >= other and self <= other.

==

Alias of eql?

hash

Returns the hash value of self.

include?(x)

Returns true if x is a element of self.

has?

contains?

Alias of include.

superset?(other)n

Returns true if self containds other. This is equivalent to other.all{|x| has?(x)}.

>=

incl?

Alias of superset?.

subset?(other)

Returns true if self is a subset of other.

<=

part_of?

Alias of subset?.

<(other)

Returns true if self is a proper subset of other.

>(other)

Returns true if other is a proper subset of self.

union(other = nil)

Returns the union of self and other. If other is omitted, returns the union of the self the set of sets.

Example:

require "finite-set"
include Algebra
p Set[0, 2, 4].cup Set[1, 3] #=> {0, 1, 2, 3, 4}
s = Set[*(0...15).to_a]
s2 = s.separate{|x| x % 2 == 0}
s3 = s.separate{|x| x % 3 == 0}
s5 = s.separate{|x| x % 5 == 0}
p Set[s2, s3, s5].union #=> {1, 7, 11, 13}

|

+

cup

Alias of union.

intersection(other = nil)

Returns the intersection of self and other. If other is omitted, returns the intersection of the self the set of sets.

Example:

require "finite-set"
include Algebra
p Set[0, 2, 4].cap(Set[4, 2, 0]) #=> {0, 2, 4}
s = Set[*(0..30).to_a]
s2 = s.separate{|x| x % 2 == 0}
s3 = s.separate{|x| x % 3 == 0}
s5 = s.separate{|x| x % 5 == 0}
p Set[s2, s3, s5].cap #=> {0, 30}

&

cap

Alias of intersection.

difference(other)

Returns the set of the elements of self which are not in other.

-

Alias of difference.

each_pair

Iterates with each two different elements of self.

Example:

require "finite-set"
include Algebra
s = Set.phi
Set[0, 1, 2].each_pair do |x, y|
  s.push [x, y]
end
p s == Set[[0, 1], [0, 2], [1, 2]] #=> true

each_member(n)

Iterates with each n different elements of self.

Example:

require "finite-set"
include Algebra
s = Set.phi
Set[0, 1, 2].each_member(2) do |x, y|
  s.push [x, y]
end
p s == Set[[0, 1], [0, 2], [1, 2]] #=> true

each_subset

Iterates over each subset of self.

Example:

require "finite-set"
include Algebra
s = Set.phi
Set[0, 1, 2].each_subset do |t|
  s.append! t
end
p s.size = 2**3 #=> true

each_non_trivial_subset

Iterates over each non trivial subset of self

power_set

Returns the set of subsets.

each_product(other)

Iterates over for each x in self and each y in other

Exameple:

require "finite-set"
include Algebra
Set[0, 1].each_prodct(Set[0, 1]) do |x, y|
  p [x, y] #=> [0,0], [0,1], [1,0], [1,1]
end

product(other)

Returns the product set of self and other. The elements are the arrays of type [x, y]. If the block is given, it returns the set which consists of the value of the block.

Example:

require "finite-set"
include Algebra
p Set[0, 1].product(Set[0, 1]) #=> {[0,0], [0,1], [1,0], [1,1]}
p Set[0, 1].product(Set[0, 1]){|x, y| x + 2*y} #=> {0, 1, 2, 3]

*

Alias of product.

equiv_class([equiv])

Returns the quotient set by the equivalent relation. The relation are given as following:

1. The evaluation of the block:

   require "finite-set"
   include Algebra
   s = Set[0, 1, 2, 3, 4, 5]
   p s.equiv_class{|a, b| (a - b) % 3 == 0} #=> {{0, 3}, {1, 4}, {2, 5}}

2. The value of the instance method call(x, y) of the parameter.

   require "finite-set"
   include Algebra
   o = Object.new
   def o.call(x, y)
     (x - y) % 3 == 0
   end
   s = Set[0, 1, 2, 3, 4, 5]
   p s.equiv_class(o) #=> {{0, 3}, {1, 4}, {2, 5}}

3. The value of method indicated Symbol.

   require "finite-set"
   include Algebra
   s = Set[0, 1, 2, 3, 4, 5]
   def q(x, y)
     (x - y) % 3 == 0
   end
   p s.equiv_class(:q) #=> {{0, 3}, {1, 4}, {2, 5}}

/

Alias of equiv_class.

to_a

Returns the array of elements. The order is indefinite.

to_ary

Alias of to_a.

sort

Returns the value of to_a.sort.

power(other)

Returns the all maps from other to self. The maps are the instances of Map.

Example:

require "finite-map"
include Algebra
a = Set[0, 1, 2, 3]
b = Set[0, 1, 2]
s = 
p( (a ** b).size )      #=> 4 ** 3 = 64
p b.surjections(a).size #=> S(3, 4) = 36
p a.injections(b).size  #=> 4P3 = 24

** power

Alias of power.

surjections(other)

Returns all surjections from other toself.

injections(other)

Returns all injections from other toself.

bijections(other)

Returns all bijections from other toself.

Enumerable

File Name:

• finite-set.rb

Methods:

any?

Returns true when the block is true for some elements. This is the alias of Enumerable#find (built-in method of ruby-1.8).

all?

Returns true when the block is true for all elements. This is defined by:

!any?{|x| !yield(x)}

(built-in method of ruby-1.8).