Show that for any sets A and B, \[A = \left( A \cap B \right) \cup \left( A - B \right)\]
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Solution
\[RHS = \left( A \cap B \right) \cup \left( A - B \right)\]
\[ \Rightarrow RHS = \left( A \cap B \right) \cup \left( A \cap B' \right)\]
\[ \Rightarrow RHS = \left[ \left( A \cap B \right) \cup A \right] \cap \left( \left( A \cap B \right) \cup B' \right)\]
\[ \Rightarrow RHS = A \cap \left[ \left( A \cup B' \right) \cap \left( B \cup B' \right) \right]\]
\[ \Rightarrow RHS = A \cap \left[ \left( A \cup B' \right) \cap U \right]\]
\[ \Rightarrow RHS = A \cap \left( A \cup B' \right)\]
\[ \Rightarrow RHS = A = LHS\]
Concept: Universal Set
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