Proof by contradiction

A proof method that assumes a claim is false and derives an impossibility, thereby establishing the claim. It is also called reductio ad absurdum.

Law of excluded middle: The classical logical principle that every proposition is true or false. In classical logic, ruling out a claim’s negation establishes the claim itself.
Irrationality of the square root of 2: The theorem that no ratio of integers equals the positive square root of 2. Assuming a reduced fraction equals the root forces both numerator and denominator to be even.
Proposition: A declarative statement that has a truth value. A proof by contradiction starts with a definite proposition whose negation can be assumed.
Proof by contrapositive: A proof of an implication that establishes its contrapositive instead. It also reasons indirectly, but proves an equivalent conditional rather than deriving an outright contradiction.
Negation: The logical operation that forms a proposition denying another proposition. The method begins by assuming the negation of the statement to be proved.
Euclid's theorem on the infinitude of primes: The theorem that infinitely many prime numbers exist. A hypothetical complete list yields a number with a prime divisor absent from the list.
Logical inference: A step in reasoning that derives a conclusion from premises. The assumed negation must lead to contradiction through valid inferential steps.
Proof by cases: A proof that establishes a conclusion separately for each possibility in an exhaustive division. It splits the argument into alternatives instead of assuming the conclusion is false.
Contradiction: A conjunction of propositions that cannot all be true at once. Deriving one exposes the impossibility of the initial assumption.
Cantor's diagonal argument: A method that constructs an item differing from every item in a proposed enumeration. It refutes a supposed complete listing by producing a missing real number.

Linked from 22 pages