Verify De Morgan’s Laws for Sets A and B
Let \[ U=\{1,2,3,4,5,6,7,8,9\} \] \[ A=\{2,4,6,8\} \] \[ B=\{2,3,5,7\} \] Verify that:
(i) \[ (A \cup B)’=A’ \cap B’ \]
(ii) \[ (A \cap B)’=A’ \cup B’ \]
Solution
(i) \[ A \cup B=\{2,3,4,5,6,7,8\} \]
\[ (A \cup B)’=\{1,9\} \]
\[ A’=\{1,3,5,7,9\} \]
\[ B’=\{1,4,6,8,9\} \]
\[ A’ \cap B’=\{1,9\} \]
Hence, \[ (A \cup B)’=A’ \cap B’ \]
(ii) \[ A \cap B=\{2\} \]
\[ (A \cap B)’=\{1,3,4,5,6,7,8,9\} \]
\[ A’ \cup B’=\{1,3,4,5,6,7,8,9\} \]
Hence, \[ (A \cap B)’=A’ \cup B’ \]