Examples of Proof by Contradiction -- How to do Mathematical Proofs (PART 7)

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This is the fifth video on a series of videos on: How to do mathematical proofs. The course is structured in such a way to make the transition from applied-style problems in mathematics (sometimes referred to as engineering mathematics) to pure mathematics much smoother.

The course will cover the basic ideas and notions of logic, such as the definition of a proposition and premise, what deductive and inductive reasoning is, and the four standard methods of proofs: direct proofs, proofs by contradiction, contrapositive arguments, and proofs by induction.

Other videos in this series:

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These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.

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KyleBroder

Thank you for this video! I really appreciate your interest in bridging the gap, so to speak, between higher level mathematics and Khan Academy style teaching (not to imply that KA isn't "higher level", but their material certainly focuses more on the more fundamental skills for solving problems, which are also necessary/helpful). So, as a nascent student of mathematics attempting to transition into the more abstract/proof based variety from the more computational style (as I described above), I find your videos quite helpful!

My question is actually relevant to these proofs by contradiction, which have always been problematic to me in practice. How might we be able to prove statements that are not of the form "A is (or is not) B"? For example, how might we go about proving a statement by the contradiction method that is of the conditional form "A implies B". Or are those two types of statements really the same, and it is really a matter of converting one statement into the other? Or is it simply a matter of negating the conditional (which I believe would be "A and not(B)" for the above example) and going from there?

Thank you for your help!

patrickharvey

For the example of sqrt(2) being irrational, why did you make the assumption that p and q to be coprime? Then isn't your proof valid only for coprime numbers? If no, why is that? If yes how can we prove it for any number?

vasileioskanaris

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