AND and OR are not two ideas but one seen in a mirror — and the mirror is negation. Deny an AND and you get an OR of denials; deny an OR and you get an AND of denials. Neither gate is primary; each is the other's reflection through ¬.
∧
AND (conjunction)
¬(A∧B) = ¬A ∨ ¬B
∨
OR (disjunction)
¬(A∨B) = ¬A ∧ ¬B
one gate, mirrored
Toggle A and B: the two right-hand columns of each law are computed independently and always agree — because the identity holds for every assignment, not most. That total agreement is what makes AND and OR a true DYAD: the same operation read from two sides of the negation, provably interchangeable. Named for Augustus De Morgan (1806–1871), who wrote the law down.
green · exhaustive proof Both laws are checked here against all four input rows — the complete truth table, so this is a proof, not a sample. The equality columns match in every row (verified in-page and node-checked); there is no assignment where they differ.
amber · the reading "AND and OR are one gate mirrored" is the honest picture of an exact theorem, but calling ¬ "the mirror" is a chosen image. The duality is real; the metaphor of reflection is the frame laid over it.
red · scope This is Boolean (two-valued) logic. The De Morgan duality generalizes (sets: complement swaps ∪/∩; it fails in some many-valued/quantum logics) — a clean, provable instance of duality, not a claim every logic is self-mirroring.
DYAS · the De Morgan Mirror · ¬(A∧B)=¬A∨¬B, exact · David Lee Wise / ROOT0, with AVAN · CC-BY-ND-4.0