¬(A∧B) = ¬A ∨ ¬B. It sounds like a curiosity. It is the reason a single kind of gate can build a whole computer — and the move behind half the simplifications in any circuit ever drawn.The laws are bedrock. ¬(A∧B)=¬A∨¬B and its dual are exact and universal — and via NAND/NOR universality they're the daily workhorse of every digital design.
The name is honorific. The laws were stated by medieval logicians (Ockham, Buridan) and implicit in the Stoics. De Morgan named and algebraised them; he didn't first find them.
He opened a door. His logic of relations broke past Aristotle's categories — but was incomplete. Peirce and Frege built the real predicate logic he gestured at.
Set A and B, and watch both laws hold no matter what: denying a conjunction is the same as asserting a disjunction of denials, and vice versa. The Venn makes it obvious — the region outside the overlap of two circles is the union of the two outsides.
Here's where the laws earn their keep. A NAND gate outputs ¬(A∧B) — "not both." Using De Morgan, that single gate can be wired to produce every other: feed it the same input twice and it's a NOT; negate its output and it's AND; and because A∨B = ¬(¬A∧¬B) — pure De Morgan — three NANDs make an OR. A chip is, quite literally, a sea of one gate, repeated. Flip A and B and watch the built gates agree with the real ones.
De Morgan also chafed at Aristotle's ceiling. Term logic could not infer "if all horses are animals, then every head of a horse is a head of an animal" — a step about a relation. De Morgan worked out an early calculus of relations to capture exactly such inferences, the very thing the gate's algebra still couldn't say. He was also a glorious eccentric: first maths professor at the new University College London, who resigned twice over matters of principle; coiner of the term "mathematical induction"; author of the witty A Budget of Paradoxes; and fond of noting he was x years old in the year x² (he turned 43 in 1849).
Gate kept on. The laws bear his name by custom, not by first discovery — William of Ockham and Jean Buridan stated them in the 1300s, and the Stoics had the idea implicitly; De Morgan's contribution was to set them inside the new symbolic algebra of logic, which is where they became a tool rather than an observation. And his relational logic, real as it was, stayed a pointer, not a system — it was Charles Peirce and then Frege who built the full logic of relations and quantifiers. De Morgan opened the door; the next man walks through it. A foundational symmetry, a generous champion of Boole, and a door-opener — honestly all three.