Verify A∩(B Δ C) = (A∩B) Δ (A∩C)
Question:
Let \[ A=\{1,2,4,5\},\quad B=\{2,3,5,6\},\quad C=\{4,5,6,7\} \] Verify the identity: \[ A\cap(B\Delta C)=(A\cap B)\Delta(A\cap C) \]
Solution
First find the symmetric difference \( B\Delta C \):
\[ B\Delta C=(B-C)\cup(C-B) \]Now,
\[ B-C=\{2,3\} \] \[ C-B=\{4,7\} \]Therefore,
\[ B\Delta C=\{2,3,4,7\} \]Now find \( A\cap(B\Delta C) \):
\[ A\cap(B\Delta C) = \{1,2,4,5\}\cap\{2,3,4,7\} \] \[ A\cap(B\Delta C)=\{2,4\} \]Now find \( A\cap B \):
\[ A\cap B = \{1,2,4,5\}\cap\{2,3,5,6\} \] \[ A\cap B=\{2,5\} \]Next find \( A\cap C \):
\[ A\cap C = \{1,2,4,5\}\cap\{4,5,6,7\} \] \[ A\cap C=\{4,5\} \]Now find the symmetric difference:
\[ (A\cap B)\Delta(A\cap C) \]Common element in both sets is 5, so remove it.
\[ (A\cap B)\Delta(A\cap C)=\{2,4\} \]Therefore,
\[ A\cap(B\Delta C)=(A\cap B)\Delta(A\cap C) \]Hence verified.