DeMorgan's theorem problem

Thread Starter

mcc123pa

Joined Sep 12, 2010
40
Hi-

I was assigned this problem for homework:

Using DeMorgan's theorem, express the function:
F= AB'C+A'C'+AB
a) with only OR and complement operations
b) with only AND and complement operations

I solved this problem and got the following answers:

Part a:
F=AB'C+A'C'+AB
F=(AB'C)'+(A'C')'+(AB)'
F=(A'+B+C')+(A+C)+(A'B')

Part b:
F=AB'C+A'C'+AB
F=(AB'C)'+(A'C')'+(AB)'
F=(A'.B.C').(A.C).(A'.B'.) (with the period being the AND function)

Should I have double negated these at the first line?
Are these answers correct or should I place a bar at the end of each parentheses thus negating everything inside of it once again?
 

Georacer

Joined Nov 25, 2009
5,182
Sadly they are wrong. Thing of it this way:
If F=A+B+C, then certainly F isn't equivalent to F=A'+B'+C'! What you must do is this:

F=(F')' %Double negation leaves the expression as is
F=((A+B+C)')'
F=(A'B'C')'

That transforms the expression for use only with AND operators. For OR operators, just double negate the terms themselves. Am I clear?
 

Thread Starter

mcc123pa

Joined Sep 12, 2010
40
Hi-

Thanks for all of your help today so far!! I think I understand what you're saying, take a look at what I have here:

F=AB'C+A'C'+AB

part a)
F=(F')'
F=((AB'C+A'C'+AB)')
F=(A'BC'.AC.A'B')' Is this how the final answer should look? Please let me know if it is worng.

part b)
F=(F')'
F=A''+B''+A'''+C'''+A''+B''
F=A+B+A'+C'+A+B Is this how the final answer should look for part B? Please let me know if it is wrong.
 

Georacer

Joined Nov 25, 2009
5,182
Well, not exactly...

Here 's how it goes:

part a):
F=AB'C+A'C'+AB
F=((AB'C)')'+((A'C')')'+((AB)')'
...

part b):
F=((AB'C+A'C'+AB)')'
F=((AB'C)'(A'C')'(AB)')'
...
Remember, (AB)'=A'+B' and (A+B)'=(A'B'). Can you continue from where I stopped?

Also, what you wrote on part b) is totally wrong. Stick to your textbook rules and revise them frequently during your first steps of boolean algebra. It can get quite confusing.
 

Thread Starter

mcc123pa

Joined Sep 12, 2010
40
Hi:

Thanks for your response.

I am still having difficulty with this problem.

Part A:

I took it from ((AB'C)')'+((A'C')')'+((AB)')'

to (A'BC')'+(AC)'+(A'B')' and am stuck with how to procede.

Part B:
Unfortunately I am completely lost on this one.

If you are able, would you be able to maybe give the answer and try to explain how you got there? I am lost unfortunately and would greatly appreciate it if you could do this. Thanks.
 

Georacer

Joined Nov 25, 2009
5,182
OK, first of all let's clarify one thing:
I took it from ((AB'C)')'+((A'C')')'+((AB)')'

to (A'BC')'+(AC)'+(A'B')'
(AB'C)' equals A'+B+C'. Take a moment to understand why, taking on account my previous post (the "Remember" part).

So we have:

part a)
F=AB'C+A'C'+AB
F=((AB'C)')'+((A'C')')'+((AB)')'
F=(A'+B+C')'+(A+C)'+(A'+B')'

And there you have it, only OR and Complementations.

part b)
F=(F')'
F=((AB'C+A'C'+AB)')'
F=((AB'C)'(A'C')'(AB)')'

That was ready since the last post, sorry for the misconception. As you can see, it contains only AND and Complementatry operators.

Got it?
 
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