Math 2.0

Mathematical logic and foundations Theorem

De Morgan's laws

Builds on NegationConjunctionDisjunctionLogical equivalence

Statement

For all propositions pp and qq, these two logical equivalences hold. ¬(p∨q)≡¬p∧¬q\neg (p \lor q) \equiv \neg p \land \neg q ¬(p∧q)≡¬p∨¬q\neg (p \land q) \equiv \neg p \lor \neg q

Remarks

The first law is also valid in intuitionistic logic. In the second law, only the direction from ¬p∨¬q\neg p \lor \neg q to ¬(p∧q)\neg (p \land q) is. The laws for sets have the same form. See De Morgan's laws for sets. The negation of a universal statement is the analogue for quantifiers.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDe Morgan's laws, statement, no evidence levelDe Morgan's lawsstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Conjunction definition
  3. Disjunction definition
  4. Implication definition
  5. Negation definition
  6. Biconditional definition
  7. Truth table concept
  8. Propositional logic concept
  9. Tautology definition
  10. Logical equivalence definition
  11. De Morgan's laws statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/De_Morgan's_laws, lead section
About this page
Tag09PKJKA6J0
URIm2://09PKJKA6J0
Aliasesde-morgan-laws
Kindstatement, theorem
WritingW1 stated
MSC03B05
StatusThis copy of the site has no build data. Run m2 status 09PKJKA6J0.
UsesNegation, Conjunction, Disjunction, Logical equivalence
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