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P. Oxy. 29, one of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5. [1] A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the
The subject of logic, in particular proof theory, formalizes and studies the notion of formal proof. [8] In some areas of epistemology and theology, the notion of justification plays approximately the role of proof, [9] while in jurisprudence the corresponding term is evidence, [10] with "burden of proof" as a concept common to both philosophy ...
A proof by contrapositive is a direct proof of the contrapositive of a statement. [14] However, indirect methods such as proof by contradiction can also be used with contraposition, as, for example, in the proof of the irrationality of the square root of 2 .
In printed English language texts, the formal statements of theorems, lemmas, and propositions are set in italics by tradition. The beginning of a proof usually follows immediately thereafter, and is indicated by the word "proof" in boldface or italics. On the other hand, several symbolic conventions exist to indicate the end of a proof.
Mathematical proof, a convincing demonstration that some mathematical statement is necessarily true; Proof complexity, computational resources required to prove statements; Proof procedure, method for producing proofs in proof theory; Proof theory, a branch of mathematical logic that represents proofs as formal mathematical objects
Traditionally, a proof is a platform which convinces someone beyond reasonable doubt that a statement is mathematically true. Naturally, one would assume that the best way to prove the truth of something like this (B) would be to draw up a comparison with something old (A) that has already been proven as true.
More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, [2] and reductio ad ...
A proof, then, laid out in accordance with the Suppes–Lemmon notation style, [43] is a sequence of lines containing sentences, [38] where each sentence is either an assumption, or the result of applying a rule of proof to earlier sentences in the sequence. [38]