DeMorgan's Laws

DeMorgan's laws are a collection of replacement rules for propositional logic that state that the negation of the conjunction of any two propositions is logically equivalent to the disjunction of the negations of those two propositions. They also state that the negation of the disjunction of any two propositions is logically equivalent to the conjunction of the negations of those propositions In symbols:

• $\lnot (A \land B) \Leftrightarrow (\lnot A \lor \lnot B)$

• $\lnot (A \lor B) \Leftrightarrow (\lnot A \land \lnot B)$

In set theory, it is the rule that the complement of the union of any two sets is equal to the intersection of the complements of those two sets. Also, they state that the complement of the intersection of any two sets is equal to the union of the complements of those sets. In symbols:

• $(A \cup B)^c = (A^c \cap B^c)$
• $(A \cap B)^c = (A^c \cup B^c)$

References

Natural Deduction Transformation Rules

Rules of inference

Modus Ponens | Modus Tollens | Disjunctive Syllogism | Hypothetical Syllogism | Conjunction Introduction | Conjunction Elimination | Disjunction Introduction | Disjunction Elimination | Bicondional Introduction | Biconditional Elimination | Constructive Dilemma | Destructive Dilemma | Absorption | Modus ponendo tollens

Rules of Transformation

Double Negation | Associative property | Commutative property | Distributive property | DeMorgan's Laws | Tautology | Exportation | Material Implication | Transposition

Set Theory

Concepts

Set theory, Set, Element, Subset, Equality, Empty Set, Enumeration, Function, Ordered pair, Uncountable set, Extensionality, Finite set, Domain, Codomain, Image, DeMorgan's Laws

Operations

Union, Intersection, Relative complement, Absolute complement, Symmetric complement, Cartesian product, Power Set

Axiomatic Systems

ZF Set Theory