Question

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: 

\[\left( A \cup B \right)' = A' \cap B'\] 

Answer 1

Given:
U = {2, 3, 5, 7, 9}
A = {3, 7}
B = {2, 5, 7, 9} 

To prove : 

\[\left( A \cup B \right)' = A' \cap B'\] 

Proof :  

LHS: 

\[(A \cup B) = {2, 3, 5, 7, 9}\]
\[(A \cup B)' = \phi\] 

RHS: 

\[A' = {2, 5, 9}\]
\[B' = {3}\]
\[A' \cap B' = \phi\]
\[LHS = RHS\]

∴\[\left( A \cup B \right)' = A' \cap B'\]