In logic, De Morgan's laws (or De Morgan's theorem), named for nineteenth century logician and mathematician Augustus De Morgan, are two powerful rules of boolean algebra and set theory:
These can be proved simply: either carefully following the process of taking complements with a Venn diagram suffices or using a truth table like this:
p q | not(p or q) | not(p) and not(q)
+--------------+------------------ T T | F | F T F | F | F F T | F | F F F | T | Tp q | not(p and q) | not(p) or not(q)
+--------------+------------------ T T | F | F T F | T | T F T | T | T F F | T | T
This simple fact is used extensively in digital circuit design for manipulating the types of logic gates used by the circuit.
A propositional expression P(p, q, ...) depending on elementary propositions p, q, ... has a De Morgan dual in which, roughly speaking, conjunction and disjunction are interchanged. We can write it as