Formulas

Number Sets

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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 \(C = A \cup B =\) \( \left\{ {x \mid x \in A\;\text{or}\;x \in B} \right\}\)
    6. Union of sets
    7. Commutativity of union \(A \cup B = B \cup A\)
    8. Associativity of union \(A \cup \left( {B \cup C} \right) =\) \( \left( {A \cup B} \right) \cup C\)
    9. Intersection of sets \(C = A \cap B =\) \( \left\{ {x \mid x \in A\;\text{and}\;x \in B} \right\}\)
    10. Intersection of sets
    11. Commutativity of intersection \(A \cap B = B \cap A\)
    12. Associativity of intersection \(A \cap \left( {B \cap C} \right) = \left( {A \cap B} \right) \cap C\)
    13. Distributivity
      \(A \cup \left( {B \cap C} \right) =\) \( \left( {A \cup B} \right) \cap \left( {A \cup C} \right)\)
      \(A \cap \left( {B \cup C} \right) =\) \( \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\)
    14. Idempotency
      \(A \cap A = A\)
      \(A \cup A = A\)
    15. Domination (Intersection of any set with the empty set) \(A \cap \emptyset = \emptyset \)
    16. Union of any set with the universal set \(A \cup I = I\)
    17. Union of any set with the empty set \(A \cup \emptyset = A\)
    18. Intersection of any set with the universal set \(A \cap I = A\)
    19. Complement \(\overline A = \left\{ {x \in I \mid x \notin A} \right\}\)
    20. Properties of the Complement
      \(A \cup \overline A = I\)
      \(A \cap \overline A = \emptyset \)
    21. De Morgan’s laws
      \(\overline {\left( {A \cup B} \right)} = \overline A \cap \overline B \)
      \(\overline {\left( {A \cap B} \right)} = \overline A \cup \overline B \)
    22. Difference of sets \(C = B\backslash A =\) \( \left\{ {x \mid x \in B\;\text{and}\;x \notin A} \right\}\)
    23. Difference of sets
    24. \(B\backslash A = B\backslash \left( {A \cap B} \right)\)
    25. \(B\backslash A = B \cap \overline A \)
    26. Difference of a set from itself \(A\backslash A = \emptyset \)
    27. \(A\backslash B = A,\;\) \(\text{if}\;\;A \cap B = \emptyset \)
    28. \(\left( {A\backslash B} \right) \cap C =\) \( \left( {A \cap C} \right)\backslash \left( {B \cap C} \right)\)
    29. \(\overline A = I\backslash A\)
    30. Cartesian product \(C = A \times B =\) \( \left\{ {\left( {x,y} \right) \mid x \in A\;\text{and}\;y \in B} \right\}\)