Math 2.0

General and overarching topics Concept

Proof by contrapositive

Builds on ContrapositiveDirect proof

Summary

A proof by contrapositive proves an implication p→qp \to q by a direct proof of ¬q→¬p\neg q \to \neg p. The method is correct because p→qp \to q is logically equivalent to its contrapositive.

Remarks

Example: for an integer nn, if n2n^2 is even, then nn is even. The contrapositive is "if nn is odd, then n2n^2 is odd", which has a short direct proof. A proof by contradiction is different: it assumes pp and ¬q\neg q together, and it gets any contradiction.

Reading path

To understand this page from the ground up, read these 15 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 tableconceptModus ponens, concept, no evidence levelModus ponensconceptPropositional logic, concept, no evidence levelPropositional logicconceptMathematical proof, concept, no evidence levelMathematical proofconceptTautology, definition, no evidence levelTautologydefinitionDirect proof, concept, no evidence levelDirect proofconceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionContrapositive, statement, no evidence levelContrapositivestatementProof by contrapositive, concept, no evidence levelProof by contrapositi…concept
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. Modus ponens concept
  9. Propositional logic concept
  10. Mathematical proof concept
  11. Tautology definition
  12. Direct proof concept
  13. Logical equivalence definition
  14. Contrapositive statement
  15. Proof by contrapositive concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Contraposition, lead section
About this page
Tag0BB5TPGQ1N
URIm2://0BB5TPGQ1N
Aliasesproof-by-contraposition
Kindconcept
WritingW0 stub
MSC00A35
SpecializesMathematical proof
Part ofProof methods
UsesContrapositive, Direct proof
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