Let \(A, B, C \subset X\) be sets. Show that

(i) \(A \backslash B=A \cap B^{c}\);

(ii) \((A \backslash B) \backslash C=A \backslash(B \cup C)\);

(iii) \(A \backslash(B \backslash C)=(A \backslash B) \cup(A \cap C)\);

(iv) \(A \backslash(B \cap C)=(A \backslash B) \cup(A \backslash C)\);

(v) \(A \backslash(B \cup C)=(A \backslash B) \cap(A \backslash C)\);

(vi) \((A \cup B) \backslash C=(A \backslash C) \cup(B \backslash C)\).