Duality (Boolean algebra)

The duality principle of Boolean Algebra says: take any true Boolean identity, swap every AND with OR and every $0$ with $1$, and the result is also a true identity.

That’s why every law in Boolean algebra has a partner. Distributivity has two forms because each is the dual of the other:

$X(Y+Z)=XY+XZ\text{and}X+YZ=(X+Y)(X+Z)$

Same for identity ($X\cdot 1=X$ pairs with $X+0=X$), annihilation ($X\cdot 0=0$ pairs with $X+1=1$), and so on.

The duality principle is not the same as taking the complement. To dualize an expression, swap operators and constants but leave variables alone. To complement, you also negate every variable — which is what De Morgan’s Laws do.

In practice, dualism halves the work of memorizing Boolean algebra: learn one form of each law and the other comes free. It also gives a quick sanity check — if a transformation works in one form but its dual is suddenly wrong, you’ve made a mistake somewhere.