# DeMorgan's theorem problem

#### 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?

#### 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.

#### mcc123pa

Joined Sep 12, 2010
40
Hi:

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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