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Question
For any two sets A and B, prove the following:
\[A \cap \left( A \cup B \right)' = \phi\]
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Solution
\[LHS = A \cap \left( A \cup B \right)'\]
\[ = A \cap \left( A' \cap B' \right)\]
\[ = \left( A \cap A' \right) \cap \left( A \cap B' \right)\]
\[ = \left( \phi \right) \cap \left( A \cap B' \right) \]
\[ = \phi = RHS \left[ \phi \cap A = \phi \right]\]
Hence proved.
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