Question:
For set \(A = \left \{{a, b, c, d} \right \}, B = \left \{{b, d, e} \right \}, C = \left \{{a, b, e} \right \} \). Proof: 1) \(A \cap (B \setminus C) = (A \cap B) \setminus (A \cap C) \) 2) \(A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C) \).
Answer
1) We have: \(B \setminus C = \left \{{d} \right \}, A \cap B = \left \{{b, d} \right \}, A \cap C = \left \{{a, b} \right \} \)
$ \Rightarrow
(A \cap B) \setminus (A \cap C) = {d} $.
Inferred: \(A \cap (B \setminus C) = \left \{ {d} \right \}, (A \cap B) \setminus (A \cap C) = \left \{{d} \right \} \)
So : \(A \cap (B \setminus C) = (A \cap B) \setminus (A \cap C) \)
2) We have: \( (B \cap C) = \left \{{b, e} \right \}, A \setminus B = \left \{{a, c} \right \}, A \setminus C = \left \{{c, d} \right \} \)
Inferred: \(A \setminus (B \cap C) = \left \{{a, c, d} \right \}; (A \setminus B) \cup (A \setminus C) = \left \{{a, c, d} \right \} \)
So: \(A \setminus (B \cap C) = (A \setminus B) \cup (A \setminus C) \).