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:

- not (P and Q) = (not P) or (not Q)
- not (P or Q) = (not P) and (not Q)

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

- .