Duality principle

Also called: duality, dual, dual expression, principle of duality, dual of an expression

Swapping AND with OR and 0 with 1 throughout a true Boolean law gives another true law, called its dual.

Look at the laws of Boolean algebra and you will see they come in pairs: = A and = A, or = 1 and = 0. The duality principle explains why.

To form the dual of an expression:

swap every + (OR) with · (AND),
swap every 0 with 1,
leave every variable and every bar exactly as it is.

The principle: if an equation is true for all inputs, its dual is true too. So every law you prove gives you a second one for free.

Keep the original grouping. The dual of is , with brackets added so that the OR still happens before the AND, just as the AND did in the original.

Dual is not the same as complement. The complement also flips every literal (that is de morgans laws), and gives a different function. The dual just gives a related law.

Worked examples

Example

Dual versus complement

Find the dual and the complement of .

1.
Dual: swap + and ·, leave the literals: .
2.
Complement: swap + and ·, and flip every literal: .
3.
Check that the complement is right at A = 0, B = 1, C = 1: the original is 1 + 0 = 1, and = (0 + 0) · 1 = 0, its opposite.

Example

A law and its dual

Start from the absorption law = A.

1.
Swap + and ·: A · (A + B) on the left.
2.
The right side is just A, with nothing to swap.
3.
So the dual law is = A, which is also true.

Common mistakes

Flipping the literals when forming a dual. That gives the complement, not the dual.
Dropping the brackets. The dual of is , not .
Thinking an expression equals its dual. Duality relates laws; and are different functions.

Practice Duality principle

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Boolean Algebra lesson

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Duality principle is taught in Boolean Algebra.

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