Prove That (A∩B)’ = A’ ∪ B’
Question:
If \[ U=\{2,3,5,7,9\} \] is the universal set, \[ A=\{3,7\}, \quad B=\{2,5,7,9\} \] then prove that: \[ (A\cap B)’=A’\cup B’ \]
Solution
First find \( A\cap B \):
\[ A\cap B=\{7\} \]Now find \( (A\cap B)’ \):
\[ (A\cap B)’=U-(A\cap B) \] \[ (A\cap B)’=\{2,3,5,7,9\}-\{7\} \] \[ (A\cap B)’=\{2,3,5,9\} \]Now find \( A’ \):
\[ A’=U-A \] \[ A’=\{2,3,5,7,9\}-\{3,7\} \] \[ A’=\{2,5,9\} \]Next find \( B’ \):
\[ B’=U-B \] \[ B’=\{2,3,5,7,9\}-\{2,5,7,9\} \] \[ B’=\{3\} \]Now find \( A’\cup B’ \):
\[ A’\cup B’ = \{2,5,9\}\cup\{3\} \] \[ A’\cup B’=\{2,3,5,9\} \]Therefore,
\[ (A\cap B)’=A’\cup B’ \]Hence proved.