How to apply De Morgan's theorem to a boolean equation?

In summary, the author is asking for help with a boolean equation, and the student suggests that they use De Morgan's laws to simplify it.
  • #1
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Homework Statement



http://img525.imageshack.us/i/circuitg.png/

Get boolean equation and apply De Morgan.

Homework Equations


The Attempt at a Solution



I'm not sure how to do this. Treat CLK like an variable? If so, is this correct?

F = abC + AbC + ABC

I do not know. I suppsoe you then apply DeMorgan.
 
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  • #2
I think you must get a formula like X1 = ~(A and C), then simplify it using De Morgan's laws and possibly factor out C at the end.
 
  • #3
huh? I don't know what you're talking about, but that's a boolean expression. I first need to be sure I have the correct boolean expression.
 
  • #4
What is boolean algebra? It is a set of rules that allow you to manipulate boolean expressions and equations. De Morgan's laws are two of the many rules in boolean algebra. I think this question is about using De Morgan's laws to simplify a boolean expression.

The expression you found is not the formula the author of the question wanted you to find. Your formula can be simplified using boolean algebra, but it does not require De Morgan's laws to simplify. So if you want to see De Morgan's laws in action, you need another formula.

F = abC + AbC + aBC

This can also be written:

F = (~A and ~B and C) or (A and ~B and C) or (~A and B and C)

where the ~ means "not" or "= 0". The benefit of writing it this way is that De Morgan's laws can be given:

1. ~(A and B) = ~A or ~B
2. ~(A or B) = ~A and ~B

You couldn't write either of these using capital and small letters.

But your formula does not need these laws to be simplified. If you look at X1 in the circuit, it is a combination of A and CLK according to that logic gate. One might say:

X1 = ac + aC + Ac

or one might say

X1 = ~(A and C)

These are just the same. If you can see that these are the same, then consider how I found this formula. It is just the definition of a NAND gate.

And now you should be able to use the logic gates in the picture to build up the formula that the author wanted you to find, the one that needs De Morgan's laws to simplify.
 
  • #5


Yes, that is correct. To apply DeMorgan's theorem, we first need to convert the circuit into a boolean equation. In this case, the boolean equation would be:

F = (a AND b AND C) OR (NOT a AND b AND C) OR (a AND NOT b AND C)

Now, we can apply DeMorgan's theorem to this equation. According to DeMorgan's theorem, the complement of a boolean expression is obtained by interchanging AND and OR operators and complementing all the variables. So, the complement of the above equation would be:

F' = (NOT a OR NOT b OR NOT C) AND (a OR NOT b OR NOT C) AND (NOT a OR b OR NOT C)

This can be simplified further using boolean algebra rules. I will leave that as an exercise for you to try. I hope this helps!
 

1. What is Boolean algebra and how is it used?

Boolean algebra is a type of mathematical logic that deals with the truth values of variables and logical operations. It is used in computer science and engineering to design digital circuits and programming languages.

2. What are the basic logical operations in Boolean algebra?

The basic logical operations in Boolean algebra are AND, OR, and NOT. These operations are used to manipulate the truth values of variables and determine the overall truth value of an expression.

3. How do DeMorgan's laws work in Boolean algebra?

DeMorgan's laws are a set of rules that allow us to simplify complex Boolean expressions by converting them into equivalent expressions. They state that the complement of an AND expression is the same as an OR expression of the complements of the individual variables, and vice versa.

4. How is Boolean algebra different from regular algebra?

Boolean algebra differs from regular algebra in that it only deals with binary values (true or false) and has a limited set of logical operations. Regular algebra, on the other hand, works with real numbers and has a wider range of mathematical operations.

5. What are some real-world applications of Boolean algebra?

Boolean algebra has many real-world applications, such as designing digital circuits for computers and electronic devices, creating search algorithms for databases and search engines, and developing programming languages for software development.

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