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A mathematical proof is a deductive argument for a mathematical statement, ... (in #Proof by contraposition). So we can write a = 2c, where c is also an integer.
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. While some authors still use the classical abbreviation, Q.E.D., it is relatively uncommon in modern mathematical texts.
Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs. [ 3 ] Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy , in which the examination of many cases results in a probable conclusion.
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally ...
In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a ... Direct Proof from Larry W. Cusick's How To Write Proofs.
Various forms of the end-of-proof symbol In mathematics, the tombstone , halmos , end-of-proof , or Q.E.D. symbol "∎" (or " ") is a symbol used to denote the end of a proof , in place of the traditional abbreviation "Q.E.D." for the Latin phrase " quod erat demonstrandum ".
In logical argument and mathematical proof, the therefore sign, ∴, is generally used before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. While it is not generally used in formal writing, it is used in mathematics and shorthand.
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges