Reduce the following to its simplest form using laws of boolean algebra. At each step clearly state the law used for simplification.
A.B' + A'.B.C' + (A.C') + B.C

  A.B' + A'.B.C' + (A.C') + B.C
= A.B' + A'.B.C' + (A.C')(B+B') + B.C       [Complementary Law: B+B'=1]
= A.B' + A'.B.C' + A.B.C' + A.B'.C' + B.C   [Distributive Law]
= A.B' + A.B'.C' + A'.B.C' + A.B.C' + B.C   [Associative Law]
= A.B' + A'.B.C' + A.B.C' + B.C             [Absorbtion Law: A.B' + A.B'.C' = A.B']
= A.B' + B.C + B.C'(A' + A)
= A.B' + B.C + B.C'.1                       [Complementary Law: A+A'=1]
= A.B' + B(C + C')
= A.B' + B                                  [Complementary Law: C+C'=1]
= (A + B).(B' + B)                          [Distributive Law]
= A + B                                     [Complementary Law: B+B'=1]