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Question
Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that \[A \cup B\] can have.
Solution
\[\text{ We know that }n\left( A \cup B \right) = n\left( A \right) + n\left( B \right) - n\left( A \cap B \right)\]
\[ n\left( A \cup B \right) \text{ is maximum when }n\left( A \cap B \right) \text{ is minimum }\]
\[so, n\left( A \cap B \right) = 0\]
\[\text{ Hence }, n\left( A \cup B \right) = n\left( A \right) + n\left( B \right) - n\left( A \cap B \right) \]
\[ = 4 + 7 - 0\]
\[ = 11\]
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