Boolean-expression simplification F = [ AB ( C + (BC)' ) + AB' ] CD'

Question

Basing on that problem. All I have in my solution is this:

mystep1:[AB(C +(B' + C')) + AB']CD'

mystep2:[AB(CB'+ CC') + AB']CD'

mystep3: [AB(CB') + AB']CD'

mystep4:[B(A+C+B') + AB']CD'

mystep5:[AB + AC + AB'] CD'

mystep6:[AC]CD'

mystep7: ACD'

F = ACD' (my simplified answer)

Please do tell me if I have followed all the necessary rules and I have the correct answer.. I am doubtful with this. For I have tried plotting it on livewire(software for logical designs) and then tried kmapping for much more easier and faster simplification. I got different answers. I believe I have wrong plots on my logical design. But this Step-by-step solution of mine is also hanging. Really not sure.

Answer 1

Here's my in-depth solution - final answer:

Given F = [ AB ( C + (BC)' ) + AB' ] CD'

Solution:

1. [AB (C + (B' + C')) + AB'] CD'
2. [AB ((C + C') + B') + AB'] CD'
3. [AB ((1) + B') + AB'] CD'
4. [AB (1 + B') + AB'] CD'
5. [AB + AB'] CD'
6. [A(B + B')] CD'
7. [A(1)] CD'
8. [A]CD'

FINAL ANSWER: ACD'

Answer 2

You switched operations in your second step. In general, $$C+(B'+C')\ne C(B'+C')=CB'+CC'.$$ Rather, $$C+(B+C')=C+(C'+B)=(C+C')+B=1+B=1.$$ Your third, fourth, fifth, and sixth steps are also faulty, but somehow, five wrongs made a right, and you came up with the correct answer.

Despite the correct answer, and despite the fact that you'll need to take a different path from the corrected second step, I strongly recommend that you take a hard look at your third, fourth, fifth, and sixth steps to see if you can find where you made your mistakes in reasoning (so you can avoid such mistakes in the future). Let me know if you have any difficulty spotting the mistakes. (Ignore the $CD'$ on the outside, as it never comes into play in those steps.)

Answer 3

$$(AB(C + B' + C') + AB')CD'$$ but $C+C'=1$ and $1+B =1$

$$(AB + AB')CD'$$

$$A(B + B')CD'$$ $$ACD'$$