Answer to Question #316700 in Discrete Mathematics for kyara

Question

Let A, B, and C be sets. Show that (A − B) − C =

(A − C) − (B − C).

Answer 1

$\left( A-B \right) -C=\left( A\cap \bar{B} \right) -C=A\cap \bar{B}\cap \bar{C}\\\left( A-C \right) -\left( B-C \right) =\left( A\cap \bar{C} \right) -\left( B\cap \bar{C} \right) =\\=\left( A\cap \bar{C} \right) \cap \overline{B\cap \bar{C}}=A\cap \bar{C}\cap \left( \bar{B}\cup C \right) =\\=\left( A\cap \bar{C}\cap \bar{B} \right) \cup \left( A\cap \bar{C}\cap C \right) =\left( A\cap \bar{C}\cap \bar{B} \right) \cup \emptyset =A\cap \bar{B}\cap \bar{C}$