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: \[ A\cup(B\cap C)=(A\cup B)\cap(A\cup C) \]
Solution
First find \( B\cap C \):
\[ B\cap C=\{5,6\} \]Now find \( A\cup(B\cap C) \):
\[ A\cup(B\cap C) = \{1,2,4,5\}\cup\{5,6\} \] \[ A\cup(B\cap C)=\{1,2,4,5,6\} \]Now find \( A\cup B \):
\[ A\cup B = \{1,2,4,5\}\cup\{2,3,5,6\} \] \[ A\cup B=\{1,2,3,4,5,6\} \]Next find \( A\cup C \):
\[ A\cup C = \{1,2,4,5\}\cup\{4,5,6,7\} \] \[ A\cup C=\{1,2,4,5,6,7\} \]Now find \( (A\cup B)\cap(A\cup C) \):
\[ (A\cup B)\cap(A\cup C) = \{1,2,3,4,5,6\}\cap\{1,2,4,5,6,7\} \] \[ (A\cup B)\cap(A\cup C)=\{1,2,4,5,6\} \]Therefore,
\[ A\cup(B\cap C)=(A\cup B)\cap(A\cup C) \]Hence verified.