Simplify the following expressions using Boolean algebra rules and specify which rule is being used in each step.

A'(A + B) + (B + AA) (A + B) = A + B (Prove that L.H.S. = R.H.S)

A'B' + (A'+ B) (B'+ B) = A' (Prove that L.H.S. = R.H.S)

(AB(C+BD) + AB) C = BC (Prove that L.H.S. = R.H.S)

F = AB + BC (B + C) + AB'C'

a)Draw the logic diagram for the above expression for F.

b)Minimize the expression.

c)Draw the logic diagram for the reduced expression.