De Morgan's Law

By Robert Laing

What I took to calling copula notation because I read somewhere that the ∧ and ∨ symbols are called copulas (but I can’t find the reference again) is a handy mnemonic for remembering two substitution rules commonly called De Morgan’s Laws:

1. ¬(p ∧ q) can be substituted with ¬p ∨ ¬q
2. ¬(p ∨ q) can be substituted with ¬p ∧ ¬q

There’s an interesting relationship with set theory in that

1. (P ∩ Q)^c can be substituted with P^c ∪ Q^c
2. (P ∪ Q)^c can be substituted with P^c ∩ Q^c

A handy feature of De Morgan’s Law is it isn’t just binary. It applies to any number of propositions:

p₁ · p₂ · … · p_i = p₁ + p₂ + … + p_i