Discrete mathematicsLesson 1 of 2 in the course Mathematical logic and proof

Mathematical Proof: Foundations and Strategies

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A proof is more than evidence that a statement seems likely: it is a deductive argument showing that assumptions logically guarantee a conclusion. This lesson introduces the purpose of proof, the role of axioms and conjectures, and the historical development of mathematical proof.

A mathematical proof is a deductive argument that establishes a mathematical statement from stated assumptions. A proof may use definitions, axioms, and previously proved results such as theorems. In principle, its reasoning can be traced back to basic assumptions and accepted rules of inference.

All flashcards

What is a mathematical proof?
A deductive argument that shows how stated assumptions, definitions, and accepted inference rules guarantee a mathematical conclusion.
What is an axiom?
A basic assumption accepted as a starting point within a mathematical system.
How does a theorem differ from a conjecture?
A theorem has been established by proof; a conjecture is believed to be true but has not yet been proved.
Why are examples insufficient to prove a universal statement?
Examples cover only selected cases. A universal claim requires reasoning that applies to every case in its stated scope.
What did Euclid contribute to proof?
Euclid systematized the axiomatic method: he began with basic assumptions and used deductive reasoning to prove propositions.

Quiz questions

  1. What are the two essential parts of a standard proof by mathematical induction?Answer: A verified base case and a step from an arbitrary case to its successor.
  2. Which statement is the contrapositive of “if p, then q”?Answer: If not q, then not p.
  3. What is the defining move in a proof by contradiction?Answer: Assume the claim is false and derive an impossibility.
  4. What does a probabilistic proof establish when a candidate has a positive probability of having a desired property?Answer: At least one candidate with that property exists.
  5. What does it mean for a statement to be undecidable from a specified set of axioms?Answer: Neither the statement nor its negation can be proved from those axioms.

Based on the Wikipedia article Mathematical proof, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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