Verify A∩(B − C) = (A∩B) − (A∩C)
Question:
Let \[ A=\{1,2,4,5\},\quad B=\{2,3,5,6\},\quad C=\{4,5,6,7\} \] Verify the identity: \[ A\cap(B-C)=(A\cap B)-(A\cap C) \]
Solution
First find \( B-C \):
\[ B-C=\{2,3\} \]Now find \( A\cap(B-C) \):
\[ A\cap(B-C) = \{1,2,4,5\}\cap\{2,3\} \] \[ A\cap(B-C)=\{2\} \]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 \( (A\cap B)-(A\cap C) \):
\[ (A\cap B)-(A\cap C) = \{2,5\}-\{4,5\} \] \[ (A\cap B)-(A\cap C)=\{2\} \]Therefore,
\[ A\cap(B-C)=(A\cap B)-(A\cap C) \]Hence verified.