Dowód przez sprzeczność

Proof by contradiction To prove p ⇒ q we prove ( p ∧ ∼ 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, b -natural number. √ 2 = a b | 2 ⇒ 2 = a 2 b 2 ⇒ a 2 = 2 b 2 . That means a 2 - is even, so a is even, thus: ∃ k ∈ Z a = 2 k. So we have: 4 k 2 = 2 b 2 ⇒ b 2 = 2 k 2.
That means b 2 - is even, so b is even.
Thus a b is reducible, what contradicts the irreducibility.
Therefore we conclude: the assumption that √ 2 is
rational was false, so √ 2 is irrational.
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