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For Any Two Sets a and B, ( a − B ) ∪ ( B − a ) = - Mathematics

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MCQ

For any two sets A and B,\[\left( A - B \right) \cup \left( B - A \right) =\] 

Options

  • (a) \[\left( A - B \right) \cup A\] 

  • (b)\[\left( B - A \right) \cup B\]

  • (c)\[\left( A \cup B \right) - \left( A \cap B \right)\]

  • (d)\[\left( A \cup B \right) \cap \left( A \cap B \right)\]

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Solution

(c)\[\left( A \cup B \right) - \left( A \cap B \right)\]

\[\left( A - B \right) \cup \left( B - A \right) = \left( A \cap B' \right) \cup \left( B \cap A' \right)\]
\[ = \left[ A \cup \left( B \cap A' \right) \right] \cap \left[ B' \cup \left( B \cap A' \right) \right] \left[ \text{ Using distribution law } \right]\]
\[ = \left[ \left( A \cup B \right) \cap \left( A \cup A' \right) \right] \cap \left[ \left( B' \cup B \right) \cap \left( B' \cup A' \right) \right] \left[ \text{ Using distribution law } \right]\]
\[ = \left[ \left( A \cup B \right) \cap \left( U \right) \right] \cap \left[ \left( U \right) \cap \left( B' \cup A' \right) \right] \left[ A \cup A' = U = B' \cup B \right] \]
\[ = \left[ A \cup B \right] \cap \left[ B' \cup A' \right] \left[ \left( A \cup B \right) \cap \left( U \right) = \left( A \cup B \right) \text{ and } \left( U \right) \cap \left( B' \cup A' \right) = \left( B' \cup A' \right) \right] \]
\[ = \left[ A \cup B \right] \cap \left[ \left( A \cap B \right)' \right] \left[ \left( A \cap B \right)' = B' \cup A' \right]\]
\[ = \left[ A \cup B \right] \cap \left[ \left( A \cup B \right) - \left( A \cap B \right) \right]\]
\[ = \left[ \left( A \cup B \right) - \left( A \cap B \right) \right]\]

Concept: Operations on Sets - Difference of Sets
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Q 8 | Page 50

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