De Morgan's Laws
Show how negation distributes over conjunction and disjunction
¬(P∧Q) ≡ (¬P∨¬Q)De Morgan's Laws are among the most fundamental and powerful rules in logic, named after mathematician Augustus De Morgan. These laws show how negation distributes over conjunction and disjunction, effectively allowing you to 'push' negation inward or 'pull' it outward while changing the connective.
The Laws Explained:
- • Law 1: 'Not (P and Q)' is equivalent to '(Not P) or (Not Q)'
- • Law 2: 'Not (P or Q)' is equivalent to '(Not P) and (Not Q)'
- • Key insight: Negation changes AND to OR, and OR to AND
Intuitive Understanding:
Think about it naturally: 'It's not true that (it's both sunny AND warm)' means 'Either it's not sunny OR it's not warm (or both)'. Similarly, 'It's not true that (it's sunny OR it's warm)' means 'It's not sunny AND it's not warm'.
¬(P ∧ Q) ≡ (¬P ∨ ¬Q) and ¬(P ∨ Q) ≡ (¬P ∧ ¬Q)De Morgan's Law for AND
Original:
De Morgan's Form:
De Morgan's Law for OR
Original:
De Morgan's Form:
Complex Application
Complex Statement:
After De Morgan's:
Key Point: De Morgan's laws show how negation distributes over conjunction and disjunction by flipping the operator and negating each component.
Example 1: First Law(Weather conditions)
Original Statement:
De Morgan Equivalent:
Explanation: When it's not both sunny and warm, at least one condition fails - either it's cloudy, or it's cold, or both.
Example 2: Second Law(Event planning)
Original Statement:
De Morgan Equivalent:
Explanation: If the party won't happen in either location, then it won't happen indoors and it also won't happen outdoors (the party is cancelled).
Example 3: Programming Logic(Software conditions)
Original Code Logic:
De Morgan Equivalent:
Explanation: In programming, De Morgan's laws help simplify conditional logic. Denying access when a user lacks both login and permission is equivalent to denying access when they lack either requirement.