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Question
Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:
\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]
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Solution
\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]
LHS
\[(B ∆ C) = (B - C) \cup (C - B)\]
\[(B - C) = {2, 3}\]
\[(C - B) = {4, 7}\]
\[(B - C) \cup (C - B) = {2, 3, 4, 7}\]
\[ \Rightarrow (B ∆ C) = {2, 3, 4, 7}\]
\[A \cap (B ∆ C) = {2, 4}\]
RHS
\[(A \cap B) = {2, 5}\]
\[(A \cap C) = {4, 5}\]
\[(A \cap B) ∆ (A \cap C) = {(A \cap B) - (A \cap C)} \cup {(A \cap C) - (A \cap B)}\]
\[(A \cap B) - (A \cap C) = {2}\]
\[(A \cap C) - (A \cap B) = {4}\]
\[{(A \cap B) - (A \cap C)} \cup {(A \cap C) - (A \cap B)} = {2, 4}\]
\[ \Rightarrow (A \cap B) ∆ (A \cap C) = {2, 4}\]
LHS = RHS
∴ \[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]
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