# Concept of Dual!?

Discussion in 'Electronic Basics' started by Steven O., Oct 17, 2004.

1. ### Steven O.Guest

I'm just learning digital electronics, taking an introductory class.
Everything is pretty clear so far, except the concept of a "dual" of a
logical function has me slightly puzzled.

I think it's like this: If I take any true logical equation, and
reverse the operators, and interchange 1s and 0s, (and, do NOT switch
A to A' or vice-versa), the equation I get as a result is still true,
and is the dual of the original -- but the new equation is NOT the
equivalent of the orginal. Is that right?

For example:

A + A' = 1
AA' = 0

Or another example:

A + 1 = 1
A0 = 0

Or, once more:

A + 0 = A
A1 = A

Are each of these pairs, in fact, the dual of each other? Thanks in

Steve O.

"Spying On The College Of Your Choice" -- How to pick the college that is the Best Match for a high school student's needs.
www.SpyingOnTheCollegeOfYourChoice.com

2. ### RatchGuest

Duality: Every Boolean expression remains valid if the operations and
identity elements are interchanged.

Ex: If (x+y)' = x' * y' holds, then (x * y)' = x' + y' also holds.

Ex: If x + 1 = 1 holds, then x * 0 = 0

Ratch

the Best Match for a high school student's needs.

3. ### William ElliotGuest

To prove AA' = 0 from A + A' = 1 use DeMorgan
(A + A')' = 1'
A' A" = 0
A' A = 0
AA' = 0
(A + 1)' = 1'
A' 1' = 0
A' 0 = 0

but as this is for all A, we have for all A
(A')' 0 = 0
A0 = 0
(A + 0)' = A'
A' 0' = A'
A' 1 = A'

again as this is for all A,
A" 1 = A"
A1 = A

Yes, and now you know the potency of DeMorgan's rules and how
to derive the dual of any universal equation.

4. ### Michael BarrGuest

These are NOT (with one exception) the dual equations, although they
are all valid. The dual expression is gotten by also complementing
each variable. So the dual of A + 0 = A is A'1 = A'. The dual of A +
B = C is A'B' = C'. Note that the latter equation is not a tautology,
which all of your examples were. Now it is a characteristic of a
tautology that if you replace each free variable by an arbitrary
expression, you still get a tautology. So if you begin with A + 0 = A
and replace A by A', you get the tautology A' + 0 = A'. If you now
dualize, you get A1 = A, again a tautology. Now A + B = C is a
contingent expression; its truth depends on those of A, B and C, but A
+ B = C for exactly the same values as A'B' = C' and NOT those that
make AB = C true.

So to repeat, to dualize an expression (or an equation), exchange 0
and 1, exchange meet and join and complement each variable.

5. ### Steven O.Guest

Thank you.

Steve O.

"Spying On The College Of Your Choice" -- How to pick the college that is the Best Match for a high school student's needs.
www.SpyingOnTheCollegeOfYourChoice.com