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Problem: Simplify the following expression using Boolean Algebra: $$ z = (B + \overline C)(\overline B + C) + \overline{ \overline A + B + \overline C} $$
Answer:
\begin{align*} z &= (B + \overline C)(\overline B + C) + A \overline{B} C \\ z &= \overline C \, \overline B + BC + A \overline{B} C \\ z &= \overline B \, \overline C + C ( B + A \overline B ) \\ z &= \overline B \, \overline C + C ( A + B ) \\ z &= AC + BC + \overline B \, \overline C \\ \end{align*} However, the book gets: $$ BC + \overline B ( \overline C + A ) $$ I believe both answers are right but I would like to know how to get the book's answer.

Bob
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    At first glance, one thing is you didn't convert AND to OR for ABC in your answer's first line. – TonyM Sep 24 '19 at 13:35
  • @TonyM You are right. I will fix it. – Bob Sep 24 '19 at 13:39
  • @TonyM The error was in the question, not the first line of my answer. I now fixed the post. – Bob Sep 24 '19 at 13:41
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    The answers are not equivalent, and yours has an error between the 4th and 5th line. – Kevin Kruse Sep 24 '19 at 13:53
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    Consider ABC = 101 and check both expressions. – Eugene Sh. Sep 24 '19 at 14:00
  • @EugeneSh Now that I have updated by post, I claim that the original expression, my answer and the book's answer all get $1$ for the case of $ABC = 101$. – Bob Sep 24 '19 at 14:07
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    Yes, now these are equivalent – Eugene Sh. Sep 24 '19 at 14:10
  • This is an excellent example of how to ask a homework question. – StainlessSteelRat Sep 24 '19 at 15:19
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    Didn't you ask the same question yesterday – Meenie Leis Sep 24 '19 at 17:47
  • @Meenie Leis I did not ask this exact question yesterday. I did ask a similar question. This is not a homework question for me. That is, I am not in school. – Bob Sep 24 '19 at 20:06
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    You must be kidding. It's the same boolean expression. For which two different reductions were discussed – Meenie Leis Sep 25 '19 at 11:58
  • @MeenieLeis You are right. I did ask the same question twice. I did not realize that to now. I was going to delete it, but I got a warning from the system that I should not delete a question with answers so I am not going to delete it. If a moderator or some other person in authority wants to delete it, I have no problem with that. Thanks for catching the error. – Bob Sep 25 '19 at 22:49

1 Answers1

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Take your answer: $$ z = AC + BC + \overline B \, \overline C $$

Now make the following transformation: $$ z = AC(B+\overline B) + BC + \overline B \, \overline C $$ Expand: $$ z = ABC+A\overline B C + BC + \overline B \, \overline C $$ Use the redundancy rule \$X+XY=X\$ on the first and third terms:

$$ z = BC + A\overline B C + \overline B \overline C = $$ $$ = BC + \overline B(AC+\overline C) $$

Use another rule: \$X+\overline XY=X + Y\$ on the expression in parentheses: $$ z = BC + \overline B(A+\overline C) $$

And now it is exactly the book answer.

Eugene Sh.
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