Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Using Properties of Sets, Show that for Any Two Sets a and B, ( a ∪ B ) ∩ ( a ∩ B ′ ) = a - Mathematics

Using properties of sets, show that for any two sets A and B,$\left( A \cup B \right) \cap \left( A \cap B' \right) = A$

#### Solution

$LHS = \left( A \cup B \right) \cup \left( A \cap B' \right)$

$\Rightarrow LHS = \left\{ \left( A \cup B \right) \cap A \right\} \cup \left\{ \left( A \cup B \right) \cap B' \right\}$

$\Rightarrow LHS = \left\{ \left( A \cup B \right) \cap A \right\} \cup \left\{ \left( A \cup B \right) \cap B' \right\}$

$\Rightarrow LHS = A \cup \left\{ \left( A \cup B \right) \cap B' \right\}$

$\Rightarrow LHS = A \cup \left\{ \left( A \cap B' \right) \cup \left( B \cap B' \right) \right\} \left( \because B \cap B = \phi \right)$

$\Rightarrow LHS = A \cup \left( A \cap B' \right)$

$\Rightarrow LHS = A = RHS$

Concept: Universal Set
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.6 | Q 11 | Page 27