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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

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

\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]

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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

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