## Quick Reference

A direct proof of a statement *q* is a logically correct argument establishing the truth of *q*. A proof by contradiction assumes that *q* is false and derives the truth of some statement *r* and of its negation ¬*r*. This contradiction shows that the initial assumption cannot hold, hence establishing the truth of *q*. A more complicated example is a proof that ‘if *p* then *q*’. A proof by contradiction assumes that *p* is true and that *q* is false, and derives the truth of some statement *r* and of its negation ¬*r*. This contradiction shows that the initial assumptions cannot both hold, and so a valid proof has been given that, if *p* is true, then *q* is true.

*Subjects:*
Mathematics.

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