# Using DeMorgan's Law, write an expression for the complement of F

• Sep 23rd 2013, 11:45 AM
lamentofking
Using DeMorgan's Law, write an expression for the complement of F
Using DeMorgan's Law, write an expression for the complement of F if F(x,y,z)= xy + x'z + yz'

I know this property:

(x'y') = (x' + y') and (x' + y') = x'y'

But there's only one variable that is NOT'ed at a time, (x in the x'z and y in yz').
• Sep 23rd 2013, 01:18 PM
emakarov
Re: Using DeMorgan's Law, write an expression for the complement of F
Quote:

Originally Posted by lamentofking
I know this property:

(x'y') = (x' + y') and (x' + y') = x'y'

No, it's (xy)' = (x' + y') and (x + y)' = x'y'. The left-hand side is the negation of a conjunction or a disjunction.

Quote:

Originally Posted by lamentofking
But there's only one variable that is NOT'ed at a time, (x in the x'z and y in yz').

You don't need to transform F(x,y,z); you need to transform its negation. F is a sum (disjunction), so it's negation can be rewritten by De Morgan's law.
• Sep 24th 2013, 07:12 AM
Hartlw
Re: Using DeMorgan's Law, write an expression for the complement of F
To begin with, emakarov solved the problem.

P^Q: (and), PvQ: (or) have truth table definitions for P,Q True or False.

T’=F and F’ =T by definition.

From the truth table definitions:

(P^Q)’=P’vQ'
(PvQ)’=P’^Q’

By definition, in this context, • is ^ and + is v

Reference wiki “DeMorgans Laws”
• Sep 24th 2013, 04:24 PM
lamentofking
Re: Using DeMorgan's Law, write an expression for the complement of F
So if I negate F then F'(x,y,z)= (xy)' + xz' + y'z ?
• Sep 24th 2013, 10:39 PM
emakarov
Re: Using DeMorgan's Law, write an expression for the complement of F
No, (xy + x'z + yz')' = (xy)'(x'z)'(yz')' = (x' + y')(x'' + z')(y' + z'') = (x' + y')(x + z')(y' + z).