Prove That A − (A − B) = A ∩ B
Question:
For any two sets \( A \) and \( B \), prove that:
\[ A-(A-B)=A\cap B \]Solution
\[ A-(A-B) \] \[ =A\cap(A-B)’ \] \[ =A\cap(A\cap B’)’ \] \[ =A\cap(A’\cup B) \] \[ =(A\cap A’)\cup(A\cap B) \] \[ =\phi\cup(A\cap B) \] \[ =A\cap B \]Hence proved.