Understanding Duality in Boolean Algebra

Question

The dual of every true statement is true.

Though this is simple, I am having difficulties understanding this. For example, $A + A = A$ and $A . A = A$ are dual; and indeed they are both true.

But let us take the statement $A = 1$. Obviously, this is not a true statement ($A$ can be $0$). However, it is not false when $A = 1$.

Though the dual of $A = 1$ happens to be $A = 0$ which is false when $A = 1$. Why is that? What does it mean?

My guess: Identities of Boolean Algebra remain Identities after the dual operation. On the other hand, Equations that where true for some subset of {0, 1} may not remain true for the same subset after the dual operation.

You could resolve this by dualizing the whole statement "$A = 1$ when $A = 1$", not just part of it. — Arthur, Aug 01 '18 at 06:47
See Duality principle in boolean algebra and see Duality principle (Boolean algebra). — Mauro ALLEGRANZA, Aug 01 '18 at 06:52

score 1 · Answer 1 · answered Apr 07 '24 at 19:12

Since there's no clear answer on Stack Overflow or in my textbook, I spent two hours thinking about this yesterday and today.

Duality is most pertinent for statements that are valid, rather than those that are merely satisfiable.

Take the well-known property XY + XZ = X(Y+Z). This statement is valid, meaning it is true for all possible X, Y, Z.

Since it is valid, it follows that the corresponding dual statement, (X + Y)(X + Z) = X + YZ, is also valid.

A = 1 is not valid, it but it is satisfiable. Duality holds only over the constant values of A that make the statement true -- and in that case, the constant values would be inverted in the dual statement as well, as @Arthur explained five years ago.

Understanding Duality in Boolean Algebra

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