Verify A − (B∪C) = (A − B)∩(A − C)

Solution

First find \( B\cup C \):

\[ B\cup C=\{2,3,4,5,6,7\} \]

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

\[ A-(B\cup C) = \{1,2,4,5\}-\{2,3,4,5,6,7\} \] \[ A-(B\cup C)=\{1\} \]

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

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

Therefore,

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

Hence verified.