Let a = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the Following Identitie: a ∩ ( B δ C ) = ( a ∩ B ) δ ( a ∩ C ) - Mathematics

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:

$A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)$

Solution

$A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)$

LHS

$(B ∆ C) = (B - C) \cup (C - B)$
$(B - C) = {2, 3}$
$(C - B) = {4, 7}$
$(B - C) \cup (C - B) = {2, 3, 4, 7}$
$\Rightarrow (B ∆ C) = {2, 3, 4, 7}$
$A \cap (B ∆ C) = {2, 4}$

RHS

$(A \cap B) = {2, 5}$
$(A \cap C) = {4, 5}$
$(A \cap B) ∆ (A \cap C) = {(A \cap B) - (A \cap C)} \cup {(A \cap C) - (A \cap B)}$
$(A \cap B) - (A \cap C) = {2}$
$(A \cap C) - (A \cap B) = {4}$
${(A \cap B) - (A \cap C)} \cup {(A \cap C) - (A \cap B)} = {2, 4}$
$\Rightarrow (A \cap B) ∆ (A \cap C) = {2, 4}$

LHS = RHS
∴  $A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)$

Concept: Operations on Sets - Union Set
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 1 Sets
Exercise 1.6 | Q 2.6 | Page 27