Simplest Logic Circuit for z=0 with Inputs a, b, c

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Discussion Overview

The discussion revolves around finding the simplest logic circuit for the output ##z## based on the inputs ##a, b, c##, specifically when ##z = 0## for the combinations ##abc = 010## and ##abc = 110##. The participants explore different methods to derive the logic circuit, including algebraic simplification and the use of Karnaugh maps.

Discussion Character

  • Homework-related
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents an initial attempt at creating a logic circuit but notes that input ##a## seems irrelevant, raising questions about the problem's requirements.
  • Another participant suggests using a Karnaugh map to verify the solution, indicating they have a different answer.
  • Several participants express uncertainty about the correctness of the initial solution and the role of input ##a## in determining the output.
  • A later reply proposes an alternative expression for ##z## and suggests that both approaches will yield the same result, although it does not clarify whether this is agreed upon by all participants.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the initial solution, with some expressing doubts and others providing alternative expressions. The discussion remains unresolved regarding the simplest logic circuit.

Contextual Notes

There is uncertainty regarding the role of input ##a## in the logic circuit, as well as the potential for different interpretations of the problem statement. The discussion reflects a reliance on various methods for simplification without a definitive resolution.

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Homework Statement



Draw the simplest logic gate or circuit for the description below. The inputs are ##a, b, c## and the output is ##z##.

##z = 0## only if ##abc = 010## or ##abc = 110##

Homework Equations

The Attempt at a Solution



So i drew up a quick table, AND and OR'ed the required terms together and came up with this:

IMG_0503.jpg


Is this okay? The input ##a## seems to have disappeared, which makes no sense to me. Unless its some sort of trick question where I only need 2 inputs since the output doesn't rely on ##a## at all.
 
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Have you tried checking your answer with a Karnaugh map? I get a different answer...
 
Also, have you tried plugging the two values back into your answer...?
 
I'm trying to do this without a K-Map.

Plugging the values of ##b## and ##c## into my answer does provide the correct output, but I am unsure about the input ##a##.

I don't believe the answer is correct, and I am unsure how to algebraically simplify this if my above attempt is incorrect.
 
My apolgies! I missed in your OP that z=0 for those two terms. Your answer does look correct. Sorry for any confusion I caused...
 
berkeman said:
My apolgies! I missed in your OP that z=0 for those two terms. Your answer does look correct. Sorry for any confusion I caused...

Thank you for the clear up. I was just slightly confused about the inputs, but the logic worked out that way I guess.
 
Zondrina said:
z= 0 only if abc=010abc = 010 or abc=110
You did well. An alternative is to write ##\bar z\ =\ \bar a b \bar c\ +\ ab\bar c##

Then simplify this as you did with your expression earlier, to get z = ...

The answers will be the same. :)
 

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