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

# Show that for Any Sets a and B, a = ( a ∩ B ) ∪ ( a − B ) - Mathematics

Show that for any sets A and B, $A = \left( A \cap B \right) \cup \left( A - B \right)$

#### Solution

$RHS = \left( A \cap B \right) \cup \left( A - B \right)$
$\Rightarrow RHS = \left( A \cap B \right) \cup \left( A \cap B' \right)$
$\Rightarrow RHS = \left[ \left( A \cap B \right) \cup A \right] \cap \left( \left( A \cap B \right) \cup B' \right)$
$\Rightarrow RHS = A \cap \left[ \left( A \cup B' \right) \cap \left( B \cup B' \right) \right]$
$\Rightarrow RHS = A \cap \left[ \left( A \cup B' \right) \cap U \right]$
$\Rightarrow RHS = A \cap \left( A \cup B' \right)$
$\Rightarrow RHS = A = LHS$

Concept: Universal Set
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.6 | Q 14.1 | Page 27