De Morgan laws

Instances of duality principles, expressing the effect of complementation in set theory on union and intersection of sets; the analogous relationship between negation in propositional calculus and conjunction and disjunction. They were published by Augustus de Morgan (1806-1871) in 1858, in the forms "The contrary of an aggregate is the compound of the contraries of the aggregants" and "The contrary of a compound is the aggregation of the contraries of the components". Both laws were known in the 14th century to William of Ockham.

Let $A$, $B$ be sets in some universal domain $\Omega$ and $\complement$ denote complementation relative to $\Omega$. Then $$ \complement (A \cap B) = (\complement A) \cup (\complement B) $$ and $$ \complement (A \cup B) = (\complement A) \cap (\complement B) $$

Let $p$ and $q$ be propositions. $$ \neg(p \wedge q) = (\neg p) \vee (\neg q) $$ and $$ \neg(p \vee q) = (\neg p) \wedge (\neg q) $$

A de Morgan algebra is an abstract algebra with binary operations $\wedge,\vee$ and an involution satisfying the analogous relations.

References

• Augustus de Morgan, Trans. Cambridge Philos. Soc. 10 (1858) 208
• Donald E. Knuth, The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1, Addison-Wesley (2014) ISBN 0133488853