Discrete Mathematics: Set Differences, Unions, and Intersections

Set Differences, Unions, and Intersections


Prove each of the following statements.


  1. $$ A \setminus (B \setminus C) = (A \setminus B) \cup (A \cap C) $$


  2. $$ A \cup B =(A \setminus B) \cup B $$


  3. $$ A \cap B = A \setminus (A \setminus B) $$


  4. $$ A \setminus B = A \setminus (A \cap B) $$


  5. $$ A \cap (B \setminus C) = (A \cap B) \setminus C = (A \setminus C) \cap (B \setminus C) $$


  6. $$ (A \cup B) \setminus C = (A \setminus C) \cup (B \setminus C) $$


  7. $$ (A \setminus B) \setminus C = A \setminus (B \cup C) = (A \setminus B) \cap (A \setminus C) $$


    8. Hint: Use DeMorgan's Laws.


  8. $$ A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C) $$


    9. Hint: Use DeMorgan's Laws.


  9. $$ (A \cap B) \cup C = (A \cup C) \cap (B \cup C) $$


  10. $$ (A \cup B) \cap C = (A \cap C) \cup (B \cap C) $$

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