Understanding Demorgan's Theorem for Logic Gates

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Discussion Overview

The discussion revolves around DeMorgan's Theorem as it applies to logic gates, specifically the relationship between AND and OR operations when negation is involved. Participants explore the theorem's implications in logic design and its application in technology mapping.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the necessity of using DeMorgan's Theorem with logic gates, questioning the relationship between AND and OR operations when negated.
  • Another participant clarifies that DeMorgan's Theorem shows how negation distributes over conjunction and disjunction, providing an example to illustrate this point.
  • A different participant suggests that understanding the operation of logic gates may require restating their definitions, but expresses uncertainty about the relevance of this approach.
  • One participant mentions that applying DeMorgan's Theorem can restructure logic to meet specific requirements, particularly in the context of technology mapping to NAND or NOR gates.
  • Another participant attempts to prove DeMorgan's Law through a series of logical steps, inviting feedback on their reasoning.

Areas of Agreement / Disagreement

Participants exhibit varying levels of understanding and interpretation of DeMorgan's Theorem, with some expressing confusion while others provide clarifications. There is no consensus on the initial participant's question, and multiple interpretations of the theorem's application remain.

Contextual Notes

Some participants' contributions reflect uncertainty regarding the definitions and operations of logic gates, as well as the application of DeMorgan's Theorem in practical scenarios. The discussion includes attempts to prove the theorem, which may involve unresolved mathematical steps.

Who May Find This Useful

This discussion may be useful for individuals interested in digital logic design, computer engineering, or those seeking to understand the application of logical operations in technology mapping.

hell18
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when using Demorgan's Theorem i didn't understand why it had to be used as logic gates AND or OR?.

e.g.

not x and y not x or not y


in other words when facing a situation with 2 logic gates the same you make them the opposite of each other to get the answer?
 
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Your post is difficult to understand. DeMorgans shows how the negation distributes over the conjunction and disjunction.
For example
~(A & B) <--> (~A V ~B)
This makes sense. The left hand side says
"It is not the case that both A and B are true."
The right hand side says,
"Either A is not true or B is not true."
Clearly these say the same thing since if A and B aren't both true, then one of them has to be false. And if one of either A or B is false, then they can't both be true.
 
Well I'm not quite sure what you are asking either, but maybe I can help.

when using Demorgan's Theorem i didn't understand why it had to be used as logic gates AND or OR?.

The only answer I can come up with would be to restate the definition of these gates, and/or restate how they operate. I don't really see the point of this so I will move on.


in other words when facing a situation with 2 logic gates the same you make them the opposite of each other to get the answer?

Well, I'm not sure what you mean by "the answer", but by applying DeMorgan's theorem you will have restructured your logic to meet whatever requirements were initially set out. If you are using a technique called "Technology Mapping" (which in most cases is used to convert all your logic to NAND or NOR gates) then DeMorgan's is the theorem you would apply. Usually these gates are faster and hence the reason you might be looking for different logic.
 
Funny, I was just thinking of this on the way home and then used GOOGLE and found this

site.

Some of the folks, on the august site, were having some problems with this, so I thought it

through and here is what I concluded. Anything I did incorrectly, please advise!

Proving DeMorgan's Law:

To Prove: (AB)'=A'+B'

From AB, then (AB)' will be TRUE for all other combination of AB

i.e.: A'B+AB'+A'B' are TRUE.

A'B+AB'+A'B'= A'B+AB'+A'B'+A'B'=B'(A+A')+A'(B+B')=A'+B'

General Expansion Case:

(ABC)'=A'+D' [SETTING BC=D]=A'+(BC)'=A'+B'+C'
 

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