Converting between Sums of Products & Products of Sums

  • Thread starter bphysics
  • Start date
  • Tags
    Sums
In summary, converting from Sum of Products to Products of Sums requires identifying false cases and writing out their maxterms. These are then negated and the laws of DeMorgan are used to simplify the resulting equation. However, with the additional rules provided, this method may not be necessary.
  • #1
bphysics
35
0

Homework Statement



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

Homework Equations


DeMorgan's theorem?
(x + y)' = x'y'
(xy)' = x' + y'

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?
 
Physics news on Phys.org
  • #2
bphysics said:

Homework Statement



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

Homework Equations


DeMorgan's theorem?
(x + y)' = x'y'
(xy)' = x' + y'

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.:)
 
  • #3
I like Serena said:
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?
 
  • #4
bphysics said:
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.
 

What is the difference between a Sum of Products and a Product of Sums?

A Sum of Products is a Boolean expression in which the variables are combined using the OR operation and then the resulting terms are combined using the AND operation. A Product of Sums is a Boolean expression in which the variables are combined using the AND operation and then the resulting terms are combined using the OR operation.

When would I use a Sum of Products versus a Product of Sums?

A Sum of Products expression is typically used when the desired output is 1 for specific combinations of input variables. A Product of Sums expression is typically used when the desired output is 0 for specific combinations of input variables.

What are the steps for converting a Sum of Products to a Product of Sums?

The steps for converting a Sum of Products expression to a Product of Sums expression are as follows: distribute the terms using De Morgan's laws, use the distributive property to combine the terms, and then simplify the resulting expression.

What are the steps for converting a Product of Sums to a Sum of Products?

The steps for converting a Product of Sums expression to a Sum of Products expression are as follows: distribute the terms using De Morgan's laws, use the distributive property to combine the terms, and then simplify the resulting expression.

Can a Sum of Products and a Product of Sums expression be equivalent?

Yes, a Sum of Products and a Product of Sums expression can be equivalent if they produce the same output for all possible combinations of input variables. This means that they can be used interchangeably in a logical circuit or equation.

Similar threads

  • Calculus and Beyond Homework Help
Replies
31
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
951
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
765
Back
Top