Verify A − (B∩C) = (A − B)∪(A − C)

Solution

First find \( B\cap C \):

\[ B\cap C=\{5,6\} \]

Now find \( A-(B\cap C) \):

\[ A-(B\cap C) = \{1,2,4,5\}-\{5,6\} \] \[ A-(B\cap C)=\{1,2,4\} \]

Now find \( A-B \):

\[ A-B = \{1,2,4,5\}-\{2,3,5,6\} \] \[ A-B=\{1,4\} \]

Next find \( A-C \):

\[ A-C = \{1,2,4,5\}-\{4,5,6,7\} \] \[ A-C=\{1,2\} \]

Now find \( (A-B)\cup(A-C) \):

\[ (A-B)\cup(A-C) = \{1,4\}\cup\{1,2\} \] \[ (A-B)\cup(A-C)=\{1,2,4\} \]

Therefore,

\[ A-(B\cap C)=(A-B)\cup(A-C) \]

Hence verified.