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Theorem (Rules For Set Complements)

Let $A$, $B$, and $C$ be sets. Then the following statements are true.

  1. $A\setminus A = \emptyset$
  2. $A\setminus \emptyset = A$
  3. $A\setminus(B\cap C) = (A\setminus B)\cup(A\setminus C)$
  4. $A\setminus(B\cup C) = (A\setminus B)\cap(A\setminus C)$
  5. $A\setminus(B\setminus C) = (A\setminus B)\cup(A\cap C)$