Duality of Boolean Expressions: Can They Be Compared?

In summary: But I'm assuming you want to leave it there.No, I have not been told to remove the complement outside the parentheses.
  • #1
aruwin
208
0
Can somebody check if I answer them right?

Write the dual of the following boolean expressions:

1.x’(y+ z’)+z = x'+(yz')+z

2.x(y+ z)’y’ = x+(y'z')+y'

3.xy+ y’z’+xz = (x+y)(y+z)'(x+z)
 
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  • #2
aruwin said:
x’(y+ z’)+z = x'+(yz')+z
...
 
  • #3
NascentOxygen said:
...

Oh, it should be x’(y+ z’)+z = x'+(yz')z. Now this is correct,right?
 
  • #4
aruwin said:
x(y+ z)’y’ = x+(y'z')+y'
I surmise that you have been told of a rule you can apply when taking the dual of a complemented
expression such as (y+z)'

Can you think of a way to confirm that you are applying that rule correctly?
 
  • #5
NascentOxygen said:
I surmise that you have been told of a rule you can apply when taking the dual of a complemented
expression such as (y+z)'

Can you think of a way to confirm that you are applying that rule correctly?

Yes, by drawing the truth table.
 
  • #6
Check expression #2. :wink:
 
  • #7
NascentOxygen said:
Check expression #2. :wink:

Ok, the results are not the same :( What should I do?
 
  • #8
aruwin said:
1.x’(y+ z’)+z = x'+(yz')+z

Actually, your expressions are a bit sloppy and I think you should be encouraged to observe more mathematical rigor.

You really can't use an equals sign here: x’(y+ z’)+z = x'+(yz')+z http://physicsforums.bernhardtmediall.netdna-cdn.com/images/icons/icon9.gif [Broken]
because the two expression are NOT equal, nor are they meant to be equal.

Perhaps type it as: x’(y+ z’)+z → x'+(yz')+z

Or, even clearer: (x’(y+ z’)+z)D = x'+(yz')+z
so long as the reader is clear on what the superscript D denotes. :smile:
 
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  • #9
NascentOxygen said:
Actually, your expressions are a bit sloppy and I think you should be encouraged to observe more mathematical rigor.

You really can't use an equals sign here: x’(y+ z’)+z = x'+(yz')+z http://physicsforums.bernhardtmediall.netdna-cdn.com/images/icons/icon9.gif [Broken]
because the two expression are NOT equal, nor are they meant to be equal.

Perhaps type it as: x’(y+ z’)+z → x'+(yz')+z

Or, even clearer: (x’(y+ z’)+z)D = x'+(yz')+z
so long as the reader is clear on what the superscript D denotes. :smile:

Wait a minute. So, what you're saying is that dual expressions are not necessarily equal to each other?I thought they were always equal, it's just that we change or to and and vice versa, but the outputs are always the same. So, they're not actually equal?
 
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  • #10
aruwin said:
Wait a minute. So, what you're saying is that dual expressions are not necessarily equal to each other?
They are not necessarily equal. In fact, I think you cannot make a general comparison.

I think you'll find duality is of limited usefulness to you (except for answering exam questions). But where it can be applied is if someone goes to all the trouble of simplifying a complex Boolean expression to something equivalent, then, without any further mathematical effort, you can take their result and simply swap AND ↔ OR (and also swap constants 1 ↔ 0 ) and you'll arrive at another equation which you can with certainty say is also correct and justifying it by citing the principle of duality in Boolean algebra..

e.g., if I tell you that (a + b)' = a' . b'
then without even understanding what it says you can write its DUAL and be confident that it also is a correct and valid Boolean equation, i.e., (a . b)' = a' + b'

Well, that's my understanding anyway. :smile:

Now, back to the problem at hand. I'd say unless you have been told to remove the complement outside the parentheses, you may as well leave it there,
e.g., ( x.(y+ z)’ y’ )D = x + (y . z)' + y'

Have you been told you should remove the tick outside parentheses? Of course, if you want to remove it then apply De Morgan's theorem, as always.
 

1. What is the duality principle in boolean algebra?

The duality principle in boolean algebra states that every algebraic expression in boolean algebra has a dual expression obtained by interchanging the logical operations AND and OR, and replacing 0's with 1's and 1's with 0's.

2. How is the duality principle used in simplifying boolean expressions?

The duality principle can be used to simplify boolean expressions by using the dual expression to make the simplification process more efficient. This is because some simplification rules may be easier to apply using the dual expression rather than the original expression.

3. What are the main operations used in the duality of boolean expressions?

The main operations used in the duality of boolean expressions are AND, OR, and NOT. These operations are used to represent logical conjunction, disjunction, and negation respectively.

4. How does the duality principle relate to De Morgan's laws?

The duality principle is closely related to De Morgan's laws, which state that the negation of a conjunction is equivalent to the disjunction of the negations, and the negation of a disjunction is equivalent to the conjunction of the negations. This is because De Morgan's laws can be derived from the duality principle.

5. Can the duality principle be applied to any boolean expression?

Yes, the duality principle can be applied to any boolean expression, regardless of its complexity. This is because the principle is based on the fundamental operations of boolean algebra and can be applied to any expression that uses these operations.

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