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Proof by contradiction

To prove p =>ą we prove (p A ~~q) =>q instead. That means, that to prove a theorem we assume its negation and we try to derive a contradiction.

Example: The classic example is the proof, that /2 is irrational (q).

Assume, that /2 is rational, so /2 - a b - irreducible

fraction, where a-integer, fr-natural number. /I - a b\2=?l = ca

bl =>a2 = 2b2.

That means a2 - is even, so a is even, thus:

Bk a = 2k.

So we have: 4k2 = 2b2 =Pb2 = 2k2.

That means b2 - is even, so b is even.

Thus a

b is reducible, what contradicts the irreducibility. Therefore we conclude: the assumption that J~2 is rational was false, so /2 is irrational.