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  • For all sets A, B and C, show that (A – B) ∩ (A – C) = A – (B ∪ C)

For all sets A, B and C, show that (A - B) \cap (A - C) = A - (B \cup C)
 

Answers (1)

Let x\in (A- B)\cap (A - C)

\\\Rightarrow x\in (A-B) \text{ and } x\in (A-C) \\\Rightarrow x\in A \text{ and } (x\notin B\text{ and } x\notin C) \\\Rightarrow x\in A \text{ and } (x\notin B\cup C)

So, (A - B) \cap (A - C) \subset A - (B \cup C)\qquad \qquad (i)

Let y\in A-(B\cup C)

y \in A - (B\cup C)\\ \Rightarrow y\in A\text{ and } (y\notin B\cup C) \\ \Rightarrow y \in A\text{ and } y\notin B \quad \text{and}\quad y\in A\text{ and }y\notin C\\ \Rightarrow y\in (A-B)\text{ and } y\in(A-C)\\ \Rightarrow y \in (A-B)\cap (A-C)

So, A - (B \cup C) \subset (A - B) \cap (A - C) \qquad \qquad (ii)

From (i) and (ii), we get,

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

    

 

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