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