For any two sets A and B, prove the following:
\[A - B = A \Delta\left( A \cap B \right)\]
Solution
\[LHS = A \Delta\left( A \cap B \right)\]
\[ = \left\{ A - \left( A \cap B \right) \right\} \cup \left\{ \left( A \cap B \right) - A \right\}\]
\[ = \left\{ A \cap \left( A \cap B \right)' \right\} \cup \left\{ \left( A \cap B \right) \cap A' \right\}\]
\[ = \left\{ A \cap \left( A' \cup B' \right) \right\} \cup \left\{ \left( A \cap B \right) \cap A' \right\}\]
\[ = \left\{ \left( A \cap A' \right) \cup \left( A \cap B' \right) \right\} \cup \left\{ \left( A \cap A' \right) \cap \left( B \cap A' \right) \right\}\]
\[ = \left\{ \left( \phi \right) \cup \left( A \cap B' \right) \right\} \cup \left\{ \left( \phi \right) \cap \left( B \cap A' \right) \right\}\]
\[ = \left( A \cap B' \right) \cup \left( \phi \right)\]
\[ = \left( A \cap B' \right)\]
\[ = A - B = RHS\]
