numy

defprotocol Numy.Set do
  @moduledoc """
  Set operations.

  Assuming vector-like container with floats.

  ## Examples

      iex(6)> a = Numy.Lapack.Vector.new(1..5)
      #Vector<size=5, [1.0, 2.0, 3.0, 4.0, 5.0]>
      iex(7)> b = Numy.Lapack.Vector.new(5..10)
      #Vector<size=6, [5.0, 6.0, 7.0, 8.0, 9.0, 10.0]>
      iex(8)> Numy.Set.union(a,b)
      #Vector<size=10, [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0]>
      iex(9)> Numy.Set.intersection(a,b)
      #Vector<size=1, [5.0]>
      iex(10)> Numy.Set.diff(a,b)
      #Vector<size=4, [1.0, 2.0, 3.0, 4.0]>
      iex(11)> Numy.Set.symm_diff(a,b)
      #Vector<size=9, [1.0, 2.0, 3.0, 4.0, 6.0, 7.0, 8.0, 9.0, 10.0]>
  """

  @doc """
  The union of two sets is formed by the elements that are present
  in either one of the sets, or in both.

  C = A ∪ B = {x : x ∈ A or x ∈ B}
  """
  def union(a, b)

  @doc """
  The intersection of two sets is formed only by the elements
  that are present in both sets.

  C = A ∩ B = {x : x ∈ A and x ∈ B}
  """
  def intersection(a, b)

  @doc """
  The difference of two sets is formed by the elements
  that are present in the first set, but not in the second one.
  """
  def diff(a, b)

  @doc """
  The symmetric difference of two sets is formed by the elements
  that are present in one of the sets, but not in the other.
  """
  def symm_diff(a, b)

  @doc """
  The [Jaccard index](https://en.wikipedia.org/wiki/Jaccard_index)
  (also known as similarity coefficient)
  measures similarity between finite sample sets, and is defined as
  the size of the intersection divided by the size of the union of the sample sets.

  ## Examples

      iex(1)> a = Numy.Lapack.Vector.new(1..6)
      #Vector<size=6, [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]>
      iex(2)> b = Numy.Lapack.Vector.new(5..10)
      #Vector<size=6, [5.0, 6.0, 7.0, 8.0, 9.0, 10.0]>
      iex(3)> Numy.Set.jaccard_index(a,b)
      0.2
  """
  def jaccard_index(a, b)

end