Math 2.0

General and overarching topics Concept

Direct proof

Builds on Mathematical proofImplication

Summary

A direct proof of an implication p→qp \to q assumes pp and then gets qq by a chain of valid steps. The steps use axioms, definitions and theorems that are already proved.

Remarks

Example: if nn is odd, then n2n^2 is odd. Write n=2k+1n = 2k + 1. Then n2=2(2k2+2k)+1n^2 = 2(2k^2 + 2k) + 1, which is odd. When a direct proof is hard, a proof by contrapositive or a proof by contradiction can help.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptImplication, definition, no evidence levelImplicationdefinitionModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptDirect proof, concept, no evidence levelDirect proofconcept
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. Implication definition
  3. Modus ponens concept
  4. Mathematical proof concept
  5. Direct proof concept

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_proof, lead section
About this page
Tag0NTF6DMVAP
URIm2://0NTF6DMVAP
Kindconcept
WritingW1 stated
MSC00A35
SpecializesMathematical proof
Part ofProof methods
UsesMathematical proof, Implication
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  • edge defines in: at least one definition (0 of 1)
  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)

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