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Converting between Sums of Products & Products of Sums

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bphysics
#1
Mar22-11, 03:21 AM
P: 35
1. The problem statement, all variables and given/known data

Translate the following equation into canonical S of P form:
F = (xyz' + x'w)(yz + x'z')

2. Relevant equations
DeMorgan's theorem?
(x + y)' = x'y'
(xy)' = x' + y'

3. The attempt at a solution
Converting from Sum of Products to Products of Sums requires the following:
Since I utilize a truth table, I locate the false cases. Then I write the maxterms out for these false cases. The resulting equation is then negated and DeMorgan's theorem is utilized to solve for the new equation, which represents the product of sums.

I've tried to perform this process in reverse to determine the sum of products for this equation. However, I get stuck, as the equation itself is not a "perfect" product of sums -- there are products within the sums (such as xyz').

How do I handle this? Am I just confused by something here? Some examples I see involve expansion (almost like FOILing it out). Also, its my understanding that canonical form is an unsimplified representation of the equation. Is this correct?
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I like Serena
#2
Mar22-11, 05:13 AM
HW Helper
I like Serena's Avatar
P: 6,187
Quote Quote by bphysics View Post
1. The problem statement, all variables and given/known data

Translate the following equation into canonical S of P form:
F = (xyz' + x'w)(yz + x'z')

2. Relevant equations
DeMorgan's theorem?
(x + y)' = x'y'
(xy)' = x' + y'

3. The attempt at a solution
Converting from Sum of Products to Products of Sums requires the following:
Since I utilize a truth table, I locate the false cases. Then I write the maxterms out for these false cases. The resulting equation is then negated and DeMorgan's theorem is utilized to solve for the new equation, which represents the product of sums.

I've tried to perform this process in reverse to determine the sum of products for this equation. However, I get stuck, as the equation itself is not a "perfect" product of sums -- there are products within the sums (such as xyz').

How do I handle this? Am I just confused by something here? Some examples I see involve expansion (almost like FOILing it out). Also, its my understanding that canonical form is an unsimplified representation of the equation. Is this correct?
Here's a few more rules that you can use:
1. (a+b)(c+d) = ac + ad + bc + bd
2. xx'=0
3. xx=x
4. x0=0
5. yz+z'=y+z'

Note that rule 1 converts the mix of products and sums to a sum-product.

With these you don't need truth tables.:)
bphysics
#3
Mar22-11, 02:39 PM
P: 35
Quote Quote by I like Serena View Post
Here's a few more rules that you can use:
1. (a+b)(c+d) = ac + ad + bc + bd
2. xx'=0
3. xx=x
4. x0=0
5. yz+z'=y+z'

Note that rule 1 converts the mix of products and sums to a sum-product.

With these you don't need truth tables.:)
So why is such a "complex" method required to convert from sums of products to products of sums? Why do I need to utilize DeMorgan's theorem for this conversion?

I like Serena
#4
Mar22-11, 04:33 PM
HW Helper
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P: 6,187
Converting between Sums of Products & Products of Sums

Quote Quote by bphysics View Post
So why is such a "complex" method required to convert from sums of products to products of sums? Why do I need to utilize DeMorgan's theorem for this conversion?
I can't answer why - it just is.
Btw, the rules are not just abstract rules - they have meaning.
For instance, rule 2 that I gave (xx' = 0) simply says that x cannot both be true and false.

In this particular problem I do not see how you would use the laws of DeMorgan.


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