Question

Prove that
A − (B ∪ C) = (A − B) ∩ (A − C),
for sets A, B and C.

Answer 1

Answer on Question #76359 – Math – Discrete Mathematics

Question

Prove that $A - (B \cup C) = (A - B) \cap (A - C)$, for sets $A, B$ and $C$.

Solution

$$A - (B \cup C) = A \cap \overline{B \cup C} = A \cap (\overline{B} \cap \overline{C}) = A \cap \overline{B} \cap \overline{C}.$$

Then $(A - B) \cap (A - C) = (A \cap \overline{B}) \cap (A \cap \overline{C}) = A \cap \overline{B} \cap A \cap \overline{C} = A \cap \overline{B} \cap \overline{C}$.

Hence $A - (B \cup C) = (A - B) \cap (A - C)$

Q.E.D.

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