(A·B)′ = Ā + B̄ — Boolean algebra laws
coreDe Morgan on a product
Answer
verified equivalent (both sides → A′ + B′)
Why this example is worth doing
De Morgan's first law, given here in the laws reference as one axiom among many rather than as the standalone tool it also gets. What this page adds is placement: De Morgan is the only law in the standard set that changes the operator, and that is why it is the one used to convert between AND-OR, NAND-only and NOR-only implementations. The page tabulates it beside the other axioms so that a student assembling a cheat sheet sees the whole system at once, with duals paired.
Try your own input in the Boolean laws & theorems. Every law, both duals, each with a truth-table proof you can check.
How the answer is reached
The law
(A · B)′A′ + B′— Both sides have the same truth table over the union of their variables.
Truth table
| # | A | B | F |
|---|---|---|---|
| 0 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 2 | 1 | 0 | 1 |
| 3 | 1 | 1 | 0 |