How Do You Derive the Sum Formula for a Full Adder?

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Discussion Overview

The discussion centers around deriving the sum formula for a full adder in digital logic, specifically focusing on the expression for the sum output (S). Participants explore various methods for simplifying the expression, including the use of Karnaugh maps and algebraic manipulation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant presents an initial expression for the sum output of a full adder and seeks assistance in simplifying it to the form S = (A ⊕ B) ⊕ C.
  • Another participant suggests using a Karnaugh map to aid in the reduction process and questions whether the original poster has utilized this method.
  • A participant expresses unfamiliarity with Karnaugh maps and questions the categorization of their inquiry as homework-related.
  • There is a discussion about the appropriateness of the thread's placement in the forum, with one participant arguing that all threads are fundamentally about understanding concepts, thus blurring the lines of what constitutes homework help.
  • Another participant questions the necessity of colored pencils for creating a Karnaugh map, implying that simpler tools could suffice.

Areas of Agreement / Disagreement

Participants express differing views on the classification of the discussion as homework-related. There is no consensus on the best method to derive the sum formula, as some advocate for the use of Karnaugh maps while others seek alternative algebraic approaches.

Contextual Notes

Participants have not reached a resolution on the method for deriving the sum formula, and there are unresolved questions regarding the use of Karnaugh maps versus algebraic methods. The discussion reflects varying levels of familiarity with the tools and concepts involved.

King
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Hi,
I've been trying to work out the formula for the sum for the full adder logic, however have come across a gap which I don't know how to fill.
S = (¬A.¬B.C) + (¬A.B.¬C) + (A.¬B.¬C) + (A.B.C)
S = ¬A.(¬B.C + B.¬C) + A.(¬B.¬C)
S = ¬A.(B [tex]\oplus[/tex] C) + A.( do not know what to do at this point to reach the next stage
S = (A [tex]\oplus[/tex] B) [tex]\oplus[/tex] C

Does anyone know how to get to that last statement?

Thanks for any help.
 
Last edited:
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King said:
Hi,
I've been trying to work out the formula for the sum for the full adder logic, however have come across a gap which I don't know how to fill.
S = (¬A.¬B.C) + (¬A.B.¬C) + (A.¬B.¬C) + (A.B.C)
S = ¬A.(¬B.C + B.¬C) + A.(¬B.¬C)
S = ¬A.(B [tex]\oplus[/tex] C) + A.( do not know what to do at this point to reach the next stage
S = (A [tex]\oplus[/tex] B) [tex]\oplus[/tex] C

Does anyone know how to get to that last statement?

Thanks for any help.


It's probably easiest to draw a Karnaugh map for the function, and use that to help your intuition in the reduction. Have you drawn a K-map for this yet?
 
Nope. I have never used one before.

Edit: How come this received a warning? This isn't homework or coursework. It's just studying.
 
King said:
Nope. I have never used one before.

Edit: How come this received a warning? This isn't homework or coursework. It's just studying.

Studying is coursework. Even if it's for self-study, it belongs in the Homework Help forums, and needs to follow the rules here. You can check out the Rules link at the top of the page for clarification.

The K-map is a great tool for helping you visualize logic reduction. Here's an intro to get you started -- there are lots of other learning resources for K-maps on the web:

http://en.wikipedia.org/wiki/Karnaugh_map

.
 
berkeman said:
Studying is coursework. Even if it's for self-study, it belongs in the Homework Help forums, and needs to follow the rules here. You can check out the Rules link at the top of the page for clarification.

That doesn't make sense since every thread in these forums is about help to understand something, so one could always link it to a course since there are courses for everything, and thus everything should therefore be in the Homework Help forums. But anyway...

Does anyone know how to solve this without the K-map since I don't really have colouring pencils to create this diagram, just a pen? Also, how would one solve this algebraically?
 
Why do you need coloured pencils for the Karnaugh map?
 

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