Proof

Proof (Encyclopedia of Science)

A proof is a logical argument demonstrating that a specific statement, proposition, or mathematical formula is true. It consists of a set of assumptions (also called premises) that are combined according to logical rules in order to establish a valid conclusion. This validation can take one of two forms. In a direct proof, a given conclusion can be shown to be true. In an indirect proof, a given conclusion can be shown not to be false and, therefore, presumably to be true.

Direct proofs

A direct proof begins with one or more axioms or facts. An axiom is a statement that is accepted as true without being proved. Axioms are also called postulates. Facts are statements that have been proved to be true to the general satisfaction of all mathematicians and scientists. In either case, a direct proof begins with a statement that everyone can agree with as being true. As an example, one might start a proof by saying that all healthy cows have four legs. It seems likely that all reasonable people would agree that this statement is true.

The next step in developing a proof is to develop a series of true statements based on the beginning axioms and/or facts. This series of statements is known as the argument of the proof. A key factor in any proof is to be certain that all of the statements in the argument are, in fact, true statements. If such is the...

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