Question

For any three sets A, B and C ______.

Options

• A ∩ (B − C) = (A ∩ B) ∪ (A ∩ C)

• A ∩ (B − C) = (A ∩ B) − C

• A ∪ (B − C) = (A ∪ B) ∩ (A ∪ C')

• A ∪ (B − C) = (A ∩ B) − (A ∩ C) 

Answer 1

For any three sets A, B and C A ∩ (B − C) = (A ∩ B) ∪ (A ∩ C).

Let x be any arbitrary element of A \[\cap \left( B - C \right)\]

Thus, we have,
x \[\in \left[ A \cap \left( B - C \right) \right]\]

\[\Rightarrow\] x\[\in A\] and x\[\in \left( B - C \right)\]

\[\Rightarrow x \in A and \left( x \in B and x \not\in C \right)\]
\[ \Rightarrow \left( x \in A and x \in B \right) and \left( x \in A and x \not\in C \right)\]
\[ \Rightarrow x\left( A \cap B \right) and x \not\in \left( A \cap C \right)\]
\[ \Rightarrow x \in \left[ \left( A \cap B \right) - \left( A \cap C \right) \right]\]
\[ \Rightarrow A \cap \left( B - C \right) \subseteq \left( A \cap B \right) - \left( A \cap C \right)\]
\[\text{ Similarly }, \left( A \cap B \right) - \left( A \cap C \right) \subseteq A \cap \left( B - C \right)\]
\[\text{ Hence }, A \cap \left( B - C \right) = \left( A \cap B \right) - \left( A \cap C \right)\]