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

Question

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

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

Sum

Solution

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

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

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

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

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

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

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

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

\[ = A - B = RHS\]

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Classification of Sets - Universal Set

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Q 2.4 Q 2.3 Q 3

Chapter 1: Sets - Exercise 1.07 [Page 34]

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R.D. Sharma Mathematics [English] Class 11

Chapter 1 Sets
Exercise 1.07 | Q 2.4 | Page 34

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