Formulas and Tables

Number Sets

Set Identities

Sets: \(A\), \(B\), \(C\)
Universal set: \(I\)
Complement: \(\overline A \)
Proper subset: \(A \subset B\)

Empty set: \(\emptyset \)
Union of sets: \(A \cup B\)
Intersection of sets: \(A \cap B\)
Difference of sets: \(A\backslash B\)

  1. \(A \subset I\) 
  2. \(A \subset A\) 
  3. \(A = B,\) if \(A \subset B\) and \(B \subset A\)
  4. Empty set \(\emptyset \subset A\)
  5. Union of sets \(C = A \cup B =\) \( \left\{ {x \mid x \in A\;\text{or}\;x \in B} \right\}\)
Union of sets
  1. Commutativity of union \(A \cup B = B \cup A\)
  2. Associativity of union \(A \cup \left( {B \cup C} \right) =\) \( \left( {A \cup B} \right) \cup C\)
  3. Intersection of sets \(C = A \cap B =\) \( \left\{ {x \mid x \in A\;\text{and}\;x \in B} \right\}\)
Intersection of sets
  1. Commutativity of intersection \(A \cap B = B \cap A\)
  2. Associativity of intersection \(A \cap \left( {B \cap C} \right) = \left( {A \cap B} \right) \cap C\)
  3. Distributivity
    \(A \cup \left( {B \cap C} \right) =\) \( \left( {A \cup B} \right) \cap \left( {A \cup C} \right)\)
    \(A \cap \left( {B \cup C} \right) =\) \( \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\)
  4. Idempotency
    \(A \cap A = A\)
    \(A \cup A = A\)
  5. Domination (Intersection of any set with the empty set) \(A \cap \emptyset = \emptyset \)
  6. Union of any set with the universal set \(A \cup I = I\)
  7. Union of any set with the empty set \(A \cup \emptyset = A\)
  8. Intersection of any set with the universal set \(A \cap I = A\)
  9. Complement \(\overline A = \left\{ {x \in I \mid x \notin A} \right\}\)
  10. Properties of the Complement
    \(A \cup \overline A = I\)
    \(A \cap \overline A = \emptyset \)
  11. De Morgan’s laws
    \(\overline {\left( {A \cup B} \right)} = \overline A \cap \overline B \)
    \(\overline {\left( {A \cap B} \right)} = \overline A \cup \overline B \)
  12. Difference of sets \(C = B\backslash A =\) \( \left\{ {x \mid x \in B\;\text{and}\;x \notin A} \right\}\)
Difference of sets
  1. \(B\backslash A = B\backslash \left( {A \cap B} \right)\)
  2. \(B\backslash A = B \cap \overline A \)
  3. Difference of a set from itself \(A\backslash A = \emptyset \)
  4. \(A\backslash B = A,\;\) \(\text{if}\;\;A \cap B = \emptyset \)
  5. \(\left( {A\backslash B} \right) \cap C =\) \( \left( {A \cap C} \right)\backslash \left( {B \cap C} \right)\)
  6. \(\overline A = I\backslash A\)
  7. Cartesian product \(C = A \times B =\) \( \left\{ {\left( {x,y} \right) \mid x \in A\;\text{and}\;y \in B} \right\}\)