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Question
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
\[\left( A \cap B \right)' = A'B' .\]
Solution
Given:
U = {2, 3, 5, 7, 9}
A = {3, 7}
B = {2, 5, 7, 9}
To prove :
\[\left( A \cap B \right)' = A' \cup B'\]
Proof :
LHS:
\[(A \cap B) = {7}\]
\[(A \cap B)' = {2, 3, 5, 9}\]
RHS:
\[A' = {2, 5, 9}\]
\[B' = {3}\]
\[A' \cup B' = {2, 3, 5, 9}\]
LHS = RHS
∴ \[\left( A \cap B \right)' = A' \cup B'\]
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