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: \[ (A\cup B)’=A’\cap B’ \]
Solution
First find \( A\cup B \):
\[ A\cup B=\{2,3,5,7,9\} \]Now find \( (A\cup B)’ \):
\[ (A\cup B)’=U-(A\cup B) \] \[ (A\cup B)’=\{2,3,5,7,9\}-\{2,3,5,7,9\} \] \[ (A\cup B)’=\phi \]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 \( A’\cap B’ \):
\[ A’\cap B’ = \{2,5,9\}\cap\{3\} \] \[ A’\cap B’=\phi \]Therefore,
\[ (A\cup B)’=A’\cap B’ \]Hence proved.