Definition

TRUE = λx. λy. x
FALSE = λx. λy. y
NOT = λb. b FALSE TRUE
AND = λb1. λb2. b1 b2 FALSE
OR = λb1. λb2. b1 TRUE b2
XOR = λb1. λb2. b1 (b2 FALSE TRUE) b2

logical_operators.ex

defmodule LogicalOperators do
  def l_true(x, _y) do
    x
  end

  def l_false(_x, y) do
    y
  end

  def l_not(b) when b == true do
    l_true(false, true)
  end

  def l_not(b) when b == false do
    l_false(false, true)
  end

  def l_and(b1, b2) when b1 == true do
    l_true(b2, false)
  end

  def l_and(b1, b2) when b1 == false do
    l_false(b2, false)
  end

  def l_or(b1, b2) when b1 == true do
    l_true(true, b2)
  end

  def l_or(b1, b2) when b1 == false do
    l_false(true, b2)
  end

  def l_xor(b1, b2) when b1 == true and b2 == true do
    l_true(l_true(false, true), b2)
  end

  def l_xor(b1, b2) when b1 == true and b2 == false do
    l_true(l_false(false, true), b2)
  end

  def l_xor(b1, b2) when b1 == false and b2 == true do
    l_false(l_true(false, true), b2)
  end

  def l_xor(b1, b2) when b1 == false and b2 == false do
    l_false(l_false(false, true), b2)
  end
end

logical_operators_test.exs

defmodule LogicalOperatorsTest do
  use ExUnit.Case, async: true
  doctest LogicalOperators

  describe "LogicalOperators.l_true/2" do
    test "returns the first argument" do
      assert LogicalOperators.l_true(1, 2) == 1
    end
  end

  describe "LogicalOperators.l_false/2" do
    test "returns the second argument" do
      assert LogicalOperators.l_false(1, 2) == 2
    end
  end

  describe "LogicalOperators.l_not/1" do
    test "NOT TRUE is FALSE" do
      assert LogicalOperators.l_not(true) == false
    end

    test "NOT FALSE is TRUE" do
      assert LogicalOperators.l_not(false) == true
    end
  end

  describe "LogicalOperators.l_and/2" do
    test "TRUE AND TRUE is TRUE" do
      assert LogicalOperators.l_and(true, true) == true
    end

    test "TRUE AND FALSE is FALSE" do
      assert LogicalOperators.l_and(true, false) == false
    end

    test "FALSE AND TRUE is FALSE" do
      assert LogicalOperators.l_and(false, true) == false
    end

    test "FALSE AND FALSE is FALSE" do
      assert LogicalOperators.l_and(false, false) == false
    end
  end

  describe "LogicalOperators.l_or/2" do
    test "TRUE AND TRUE is TRUE" do
      assert LogicalOperators.l_or(true, true) == true
    end

    test "TRUE AND FALSE is TRUE" do
      assert LogicalOperators.l_or(true, false) == true
    end

    test "FALSE AND TRUE is TRUE" do
      assert LogicalOperators.l_or(false, true) == true
    end

    test "FALSE AND FALSE is FALSE" do
      assert LogicalOperators.l_or(false, false) == false
    end
  end

  describe "LogicalOperators.l_xor/2" do
    test "TRUE AND TRUE is FALSE" do
      assert LogicalOperators.l_xor(true, true) == false
    end

    test "TRUE AND FALSE is TRUE" do
      assert LogicalOperators.l_xor(true, false) == true
    end

    test "FALSE AND TRUE is TRUE" do
      assert LogicalOperators.l_xor(false, true) == true
    end

    test "FALSE AND FALSE is FALSE" do
      assert LogicalOperators.l_xor(false, false) == false
    end
  end
end