diff --git a/src/random_events/better_variables.py b/src/random_events/better_variables.py
new file mode 100644
index 0000000..0090266
--- /dev/null
+++ b/src/random_events/better_variables.py
@@ -0,0 +1,59 @@
+from dataclasses import dataclass
+
+from typing_extensions import Self, Type
+
+from .interval import Interval, SimpleInterval
+from .set import Set, SetElement
+from .sigma_algebra import AbstractCompositeSet
+
+
+@dataclass
+class Variable:
+    name: str
+    domain: AbstractCompositeSet
+
+    def __lt__(self, other: Self) -> bool:
+        """
+        Returns True if self < other, False otherwise.
+        """
+        return self.name < other.name
+
+    def __gt__(self, other: Self) -> bool:
+        """
+        Returns True if self > other, False otherwise.
+        """
+        return self.name > other.name
+
+    def __hash__(self) -> int:
+        return self.name.__hash__()
+
+    def __eq__(self, other):
+        return self.name == other.name
+
+    def __str__(self):
+        return f"{self.__class__.__name__}({self.name}, {self.domain})"
+
+    def __repr__(self):
+        return f"{self.__class__.__name__}({self.name})"
+
+
+@dataclass
+class Continuous(Variable):
+    domain: Interval = Interval([SimpleInterval(-float("inf"), float("inf"))])
+
+
+@dataclass
+class Symbolic(Variable):
+    """
+    Class for unordered, finite, discrete random variables.
+    """
+    domain: Set
+
+    def __init__(self, name: str, domain: Type):
+        super().__init__(name, Set([value for value in domain if value != domain.EMPTY_SET]))
+
+
+@dataclass
+class Integer(Variable):
+    """Class for ordered, discrete random variables."""
+    domain: Interval = Interval([SimpleInterval(-float("inf"), float("inf"))])
diff --git a/src/random_events/interval.py b/src/random_events/interval.py
index b4e7046..031c88b 100644
--- a/src/random_events/interval.py
+++ b/src/random_events/interval.py
@@ -177,3 +177,6 @@ def simplify(self) -> Self:
 
     def new_empty_set(self) -> Self:
         return Interval()
+
+    def complement_if_empty(self) -> Self:
+        return Interval([SimpleInterval(float('-inf'), float('inf'), Bound.OPEN, Bound.OPEN)])
diff --git a/src/random_events/product_algebra.py b/src/random_events/product_algebra.py
new file mode 100644
index 0000000..6d60d7c
--- /dev/null
+++ b/src/random_events/product_algebra.py
@@ -0,0 +1,96 @@
+from collections.abc import dict_keys, dict_values
+from typing import Any
+
+from sortedcontainers import SortedDict, SortedKeysView, SortedValuesView
+from typing_extensions import Union, Any
+
+from .better_variables import *
+from .sigma_algebra import *
+from .variables import Variable
+
+
+class VariableMap(SortedDict[Variable, Any]):
+    """
+    A map of variables to values.
+
+    Accessing a variable by name is also supported.
+    """
+
+    @property
+    def variables(self) -> dict_keys[Variable]:
+        return self.keys()
+
+    def variable_of(self, name: str) -> Variable:
+        """
+        Get the variable with the given name.
+        :param name: The variable's name
+        :return: The variable itself
+        """
+        variable = [variable for variable in self.variables if variable.name == name]
+        if len(variable) == 0:
+            raise KeyError(f"Variable {name} not found in event {self}")
+        return variable[0]
+
+    def __getitem__(self, item: Union[str, Variable]):
+        if isinstance(item, str):
+            item = self.variable_of(item)
+        return super().__getitem__(item)
+
+    def __setitem__(self, key: Union[str, Variable], value: Any):
+        if isinstance(key, str):
+            key = self.variable_of(key)
+
+        if not isinstance(key, Variable):
+            raise TypeError(f"Key must be a Variable if not already present, got {type(key)} instead.")
+
+        super().__setitem__(key, value)
+
+    def __copy__(self):
+        return self.__class__({variable: value for variable, value in self.items()})
+
+
+class SimpleEvent(AbstractSimpleSet, VariableMap[Variable, AbstractCompositeSet]):
+
+    @property
+    def assignments(self) -> dict_values[AbstractCompositeSet]:
+        return self.values()
+
+    def intersection_with(self, other: Self) -> Self:
+        variables = self.keys() | other.keys()
+        result = SimpleEvent()
+        for variable in variables:
+            if variable in self and variable in other:
+                result[variable] = self[variable].intersection_with(other[variable])
+            elif variable in self:
+                result[variable] = self[variable]
+            else:
+                result[variable] = other[variable]
+
+        return result
+
+    def complement(self) -> SortedSet[Self]:
+        pass
+
+    def is_empty(self) -> bool:
+
+        if len(self) == 0:
+            return True
+
+        for assignment in self.values():
+            if assignment.is_empty():
+                return True
+
+        return False
+
+    def contains(self, item) -> bool:
+        pass
+
+    def __hash__(self):
+        return hash(tuple(self.items()))
+
+    def __lt__(self, other):
+        pass
+
+
+class Event(AbstractCompositeSet):
+    ...
diff --git a/src/random_events/set.py b/src/random_events/set.py
new file mode 100644
index 0000000..e99d7ab
--- /dev/null
+++ b/src/random_events/set.py
@@ -0,0 +1,66 @@
+import enum
+from abc import abstractmethod
+
+from sortedcontainers import SortedSet
+from typing_extensions import Self
+
+from . import sigma_algebra
+
+
+class SetElement(sigma_algebra.AbstractSimpleSet, enum.Enum):
+    """
+    Base class for enums that are used as elements in a set.
+
+    Classes that inherit from this class have to define an attribute called EMPTY_SET.
+    """
+
+    @property
+    @abstractmethod
+    def EMPTY_SET(self):
+        raise NotImplementedError
+
+    @property
+    def all_elements(self):
+        return self.__class__
+
+    def intersection_with(self, other: Self) -> Self:
+        if self == other:
+            return self
+        else:
+            return self.all_elements.EMPTY_SET
+
+    def complement(self) -> SortedSet[Self]:
+        result = SortedSet()
+        for element in self.all_elements:
+            if element != self and element != self.all_elements.EMPTY_SET:
+                result.add(element)
+        return result
+
+    def is_empty(self) -> bool:
+        return self is self.all_elements.EMPTY_SET
+
+    def contains(self, item: Self) -> bool:
+        return self == item
+
+    def non_empty_to_string(self) -> str:
+        return self.name
+
+    def __hash__(self):
+        return enum.Enum.__hash__(self)
+
+    def __lt__(self, other):
+        return self.value < other.value
+
+
+class Set(sigma_algebra.AbstractCompositeSet):
+
+    simple_sets: SortedSet[SetElement]
+
+    def complement_if_empty(self) -> Self:
+        raise NotImplementedError("I don't know how to do this yet.")
+
+    def simplify(self) -> Self:
+        return self
+
+    def new_empty_set(self) -> Self:
+        return Set([])
diff --git a/src/random_events/sigma_algebra.py b/src/random_events/sigma_algebra.py
index b597e0a..e073513 100644
--- a/src/random_events/sigma_algebra.py
+++ b/src/random_events/sigma_algebra.py
@@ -163,16 +163,25 @@ def intersection_with_simple_set(self, other: AbstractSimpleSet) -> Self:
         [result.add_simple_set(simple_set.intersection_with(other)) for simple_set in self.simple_sets]
         return result
 
+    def intersection_with_simple_sets(self, other: SortedSet[AbstractSimpleSet]) -> Self:
+        """
+        Form the intersection of this object with a set of simple sets.
+
+        :param other: The set of simple sets
+        :return: The intersection of this set with the set of simple sets
+        """
+        result = self.new_empty_set()
+        [result.simple_sets.update(self.intersection_with_simple_set(other_simple_set).simple_sets)
+         for other_simple_set in other]
+        return result
+
     def intersection_with(self, other: Self) -> Self:
         """
         Form the intersection of this object with another object.
         :param other: The other set
         :return: The intersection of this set with the other set
         """
-        result = self.new_empty_set()
-        [result.simple_sets.update(self.intersection_with_simple_set(other_simple_set).simple_sets)
-         for other_simple_set in other.simple_sets]
-        return result
+        return self.intersection_with_simple_sets(other.simple_sets)
 
     def __and__(self, other):
         return self.intersection_with(other)
@@ -187,12 +196,7 @@ def difference_with_simple_set(self, other: AbstractSimpleSet) -> Self:
         [result.simple_sets.update(simple_set.difference_with(other)) for simple_set in self.simple_sets]
         return result.make_disjoint()
 
-    def difference_with(self, other: Self) -> Self:
-        """
-        Form the difference with another composite set.
-        :param other: The other set
-        :return: The difference of this set with the other set
-        """
+    def difference_with_simple_sets(self, other: SortedSet[AbstractSimpleSet]) -> Self:
 
         # initialize the result
         result = self.new_empty_set()
@@ -205,7 +209,7 @@ def difference_with(self, other: Self) -> Self:
             first_iteration = True
 
             # for every simple set in the other set
-            for other_simple_set in other.simple_sets:
+            for other_simple_set in other:
 
                 # form the element wise difference
                 difference_with_other_simple_set = own_simple_set.difference_with(other_simple_set)
@@ -227,6 +231,14 @@ def difference_with(self, other: Self) -> Self:
 
         return result.make_disjoint()
 
+    def difference_with(self, other: Self) -> Self:
+        """
+        Form the difference with another composite set.
+        :param other: The other set
+        :return: The difference of this set with the other set
+        """
+        return self.difference_with_simple_sets(other.simple_sets)
+
     def __sub__(self, other):
         return self.difference_with(other)
 
@@ -234,16 +246,23 @@ def complement(self) -> Self:
         """
         :return: The complement of this set
         """
+
+        if self.is_empty():
+            return self.complement_if_empty()
+
         result = self.new_empty_set()
-        first_iteration = True
-        for simple_set in self.simple_sets:
-            if first_iteration:
-                result = simple_set.complement()
-                first_iteration = False
-            else:
-                result = result.intersection_with(simple_set.complement())
+        result.simple_sets = self.simple_sets[0].complement()
+        for simple_set in self.simple_sets[1:]:
+            result = result.intersection_with_simple_sets(simple_set.complement())
         return result.make_disjoint()
 
+    @abstractmethod
+    def complement_if_empty(self) -> Self:
+        """
+        :return: The complement of this if it is empty.
+        """
+        raise NotImplementedError
+
     def __invert__(self):
         return self.complement()
 
diff --git a/test/test_set.py b/test/test_set.py
new file mode 100644
index 0000000..7183d1b
--- /dev/null
+++ b/test/test_set.py
@@ -0,0 +1,62 @@
+from random_events.set import SetElement, Set
+import enum
+import unittest
+
+
+class TestEnum(SetElement):
+    EMPTY_SET = 0
+    A = 1
+    B = 2
+    C = 4
+
+
+class SetElementTestCase(unittest.TestCase):
+
+    def test_intersection_with(self):
+        a = TestEnum.A
+        b = TestEnum.B
+
+        intersection_a_b = a.intersection_with(b)
+        self.assertEqual(intersection_a_b, TestEnum.EMPTY_SET)
+        self.assertEqual(a.intersection_with(TestEnum.A), a)
+        self.assertEqual(TestEnum.EMPTY_SET.intersection_with(TestEnum.A), TestEnum.EMPTY_SET)
+
+    def test_complement(self):
+        a = TestEnum.A
+        complement = a.complement()
+        self.assertEqual(complement, {TestEnum.B, TestEnum.C})
+
+    def test_is_empty(self):
+        a = TestEnum.EMPTY_SET
+        b = TestEnum.B
+        self.assertTrue(a.is_empty())
+        self.assertFalse(b.is_empty())
+
+    def test_contains(self):
+        a = TestEnum.A
+        self.assertTrue(a.contains(TestEnum.A))
+        self.assertFalse(a.contains(TestEnum.B))
+        self.assertFalse(a.contains(TestEnum.C))
+
+
+class SetTestCase(unittest.TestCase):
+
+    def test_simplify(self):
+        a = TestEnum.A
+        b = TestEnum.B
+        c = TestEnum.C
+        s = Set([a, b, c, c])
+        self.assertEqual(len(s.simple_sets), 3)
+        self.assertEqual(s.simplify(), s)
+
+    def test_difference(self):
+        s = Set([TestEnum.A, TestEnum.B])
+        s_ = Set([TestEnum.A])
+        self.assertEqual(s.difference_with(s_), Set([TestEnum.B]))
+
+    def test_complement(self):
+        s = Set([TestEnum.A, TestEnum.B])
+        self.assertEqual(s.complement(), Set([TestEnum.C]))
+
+if __name__ == '__main__':
+    unittest.main()
diff --git a/test/test_variable.py b/test/test_variable.py
new file mode 100644
index 0000000..f9afca4
--- /dev/null
+++ b/test/test_variable.py
@@ -0,0 +1,39 @@
+import unittest
+
+from random_events.better_variables import *
+from random_events.interval import *
+
+
+class TestEnum(SetElement):
+    EMPTY_SET = 0
+    A = 1
+    B = 2
+    C = 4
+
+
+class ContinuousTestCase(unittest.TestCase):
+
+    def test_creation(self):
+        x = Continuous("x")
+        self.assertEqual(x.name, "x")
+        self.assertEqual(x.domain, Interval([SimpleInterval(-float("inf"), float("inf"))]))
+
+
+class IntegerTestCase(unittest.TestCase):
+
+    def test_creation(self):
+        x = Integer("x")
+        self.assertEqual(x.name, "x")
+        self.assertEqual(x.domain, Interval([SimpleInterval(-float("inf"), float("inf"))]))
+
+
+class SymbolicTestCase(unittest.TestCase):
+
+    def test_creation(self):
+        x = Symbolic("x", TestEnum)
+        self.assertEqual(x.name, "x")
+        self.assertEqual(x.domain, Set([TestEnum.A, TestEnum.B, TestEnum.C]))
+
+
+if __name__ == '__main__':
+    unittest.main()