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For any two sets A and B, prove the following: A − B = A Δ ( A ∩ B ) - Mathematics

Sum

For any two sets A and B, prove the following:

\[A - B = A \Delta\left( A \cap B \right)\]

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Solution

\[LHS = A \Delta\left( A \cap B \right)\]

\[ = \left\{ A - \left( A \cap B \right) \right\} \cup \left\{ \left( A \cap B \right) - A \right\}\]

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

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

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

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

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

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

\[ = A - B = RHS\]

Concept: Universal Set
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.7 | Q 2.4 | Page 34
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