Let *A* = {1, 2, 4, 5} *B* = {2, 3, 5, 6} *C* = {4, 5, 6, 7}. Verify the following identitie:

\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]

#### 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

∴ \[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]