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Question
Using properties of sets, show that for any two sets A and B,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\]
Solution
\[LHS = \left( A \cup B \right) \cup \left( A \cap B' \right)\]
\[ \Rightarrow LHS = \left\{ \left( A \cup B \right) \cap A \right\} \cup \left\{ \left( A \cup B \right) \cap B' \right\}\]
\[ \Rightarrow LHS = \left\{ \left( A \cup B \right) \cap A \right\} \cup \left\{ \left( A \cup B \right) \cap B' \right\}\]
\[ \Rightarrow LHS = A \cup \left\{ \left( A \cup B \right) \cap B' \right\}\]
\[ \Rightarrow LHS = A \cup \left\{ \left( A \cap B' \right) \cup \left( B \cap B' \right) \right\} \left( \because B \cap B = \phi \right)\]
\[ \Rightarrow LHS = A \cup \left( A \cap B' \right)\]
\[ \Rightarrow LHS = A = RHS\]
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