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From The Free On-line Dictionary of Computing (05 January 2017) [foldoc]:
DeMorgan's theorem
A logical theorem which states that the complement
of a conjunction is the disjunction of the complements or
vice versa. In symbols:
not (x and y) = (not x) or (not y)
not (x or y) = (not x) and (not y)
E.g. if it is not the case that I am tall and thin then I am
either short or fat (or both). The theorem can be extended to
combinations of more than two terms in the obvious way.
The same laws also apply to sets, replacing logical complement
with set complement, conjunction ("and") with set
intersection, and disjunction ("or") with set union.
A (C) programmer might use this to re-write
if (!foo && !bar) ...
as
if (!(foo || bar)) ...
thus saving one operator application (though an {optimising
compiler} should do the same, leaving the programmer free to
use whichever form seemed clearest).
(1995-12-14)