| Mathematical Proof |
Article Index for Mathematical |
Website Links For Mathematical |
Information AboutMathematical Proof |
| CATEGORIES ABOUT MATHEMATICAL PROOF | |
| mathematical logic | |
| proofs | |
|
Proofs employ Logic but usually include some amount of Natural Language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of Informal Logic . Purely formal proofs are considered in Proof Theory . The distinction between formal and informal proofs has led to much examination of current and historical Mathematical Practice , Quasi-empiricism In Mathematics , and so-called Folk Mathematics (in both senses of that term). The Philosophy Of Mathematics is concerned with the role of language and logic in proofs, and Mathematics As A Language . Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone. Once a theorem is proved, it can be used as the basis to prove further statements. The axioms are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on Practice , i.e. acceptable techniques. METHODS OF PROOF Direct proof See Also: Direct proof In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to establish that the sum of two Even Integers is always even: :For any two even integers and we can write and for some integers and , since both and are multiples of 2. But the sum is also a multiple of 2, so it is therefore even by definition. This proof uses definition of even integers, as well as Distribution Law . Proof by induction See Also: Mathematical induction In proof by induction, first a "base case" is proved, and then an "induction rule" is used to prove a (often Infinite ) series of other cases. Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. A subset of induction is Infinite Descent . Infinite descent can be used to prove the Irrationality Of The Square Root Of Two . The principle of mathematical induction states that: Let ''N'' = { 1, 2, 3, 4, ... } be the set of natural numbers and ''P(''n'')'' be a mathematical statement involving the natural number ''n'' belonging to '''N''' such that ''(i)'' ''P''(1) is true, ie, ''P''(''n'') is true for ''n'' = 1 ''(ii)'' ''P''(''m'' + 1) is true whenever ''P''(''m'') is true, ie, ''P''(''m'') is true implies that ''P''(''m'' + 1) is true. Then ''P''(''n'') is true for the set of natural numbers ''N''. Proof by transposition See Also: Transposition (logic) Proof by Transposition establishes the conclusion "if ''p'' then ''q''" by proving the equivalent Contrapositive statement "if ''not q'' then ''not p''". Proof by contradiction See Also: Reductio ad absurdum In proof by contradiction (also known as ''reductio ad absurdum'', Latin for "reduction into the absurd"), it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true. This method is perhaps the most prevalent of mathematical proofs. A famous example of a proof by contradiction shows that is Irrational : :Suppose that is rational, so where ''a'' and ''b'' are non-zero integers with no common factor (definition of rational number). Thus, . Squaring both sides yields 2''b''2 = ''a''2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So ''a''2 is even, which implies that ''a'' must also be even. So we can write ''a'' = 2''c'', where ''c'' is also an integer. Substitution into the original equation yields 2''b''2 = (2''c'')2 = 4''c''2. Dividing both sides by 2 yields ''b''2 = 2''c''2. But then, by the same argument as before, 2 divides ''b''2, so ''b'' must be even. However, if ''a'' and ''b'' are both even, they share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that is irrational. Proof by construction See Also: Proof by construction Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville , for instance, proved the existence of Transcendental Number s by constructing an Explicit Example . Proof by exhaustion See Also: Proof by exhaustion In Proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the Four Colour Theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases. Probabilistic proof See Also: Probabilistic method A probabilistic proof is one in which an example is shown to exist by methods of Probability Theory - not an argument that a theorem is 'probably' true. The latter type of reasoning can be called a 'plausibility argument'; in the case of the Collatz Conjecture it is clear how far that is from a genuine proof. Probabilistic proof, like proof by construction, is one of many ways to show Existence Theorem s. Combinatorial proof See Also: Combinatorial proof A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways. Usually a Bijection is used to show that the two interpretations give the same result. Nonconstructive proof See Also: Nonconstructive proof A nonconstructive proof establishes that a certain mathematical object must exist (e.g. "Some X satisfies f(X)"), without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proven to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. Proof nor disproof There is a class of mathematical formulae for which neither a proof nor disproof exists, using only the standard ZFC axioms. This result is known as Gödel's (first) Incompleteness Theorem and examples include the Continuum Hypothesis . Whether a particular unproven proposition can be proved using a standard set of axioms is not always obvious, and can be extremely technical to determine. Elementary proof See Also: Elementary proof An elementary proof is (usually) a proof which does not use complex analysis. For some time it was thought that certain theorems, like the Prime Number Theorem , could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques. END OF A PROOF See Also: Q.E.D. Sometimes, the abbreviation ''"Q.E.D."'' is written to indicate the end of a proof. This abbreviation stands for ''"Quod Erat Demonstrandum"'', which is Latin for ''"that which was to be demonstrated"''. An alternative is to use a small rectangle with its shorter side horizontal (), known as a Tombstone or Halmos . SEE ALSO
REFERENCES
EXTERNAL LINKS
|
|
|