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| {{See also|Table of logic symbols}}
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| In [[logic]], '''proof by contradiction''' is a form of [[Mathematical proof|proof]] that establishes the [[Truth#Formal theories|truth]] or [[validity]] of a [[proposition]] by showing that the proposition's being false would imply a [[contradiction]]. Proof by contradiction is also known as '''indirect proof''', '''apagogical argument''', '''proof by assuming the opposite''', and '''''reductio ad impossibilem'''''. It is a particular kind of the more general form of argument known as ''[[reductio ad absurdum]]''.
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| [[G. H. Hardy]] described proof by contradiction as "one of a mathematician's finest weapons", saying "It is a far finer gambit than any [[chess]] [[gambit]]: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."<ref>[[G. H. Hardy]], ''[[A Mathematician's Apology]]; Cambridge University Press, 1992. ISBN 9780521427067. ''[http://books.google.com/books?id=beImvXUGD-MC&pg=PA94 p. 94].</ref>
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| == Examples ==
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| ===Irrationality of the square root of 2===
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| A classic proof by contradiction from mathematics is the [[Square root of 2#Proof by infinite descent|proof that the square root of 2 is irrational]].<ref>{{cite web|url=http://www.math.utah.edu/~pa/math/q1.html|title=Why is the square root of 2 irrational?|last=Alfield|first=Peter|date=16 August 1996|work=Understanding Mathematics, a study guide|publisher=Department of Mathematics, University of Utah|accessdate=6 February 2013}}</ref> If it were [[rational number|rational]], it could be expressed as a fraction ''a''/''b'' in [[lowest terms]], where ''a'' and ''b'' are [[integers]], at least one of which is [[odd number|odd]]. But if ''a''/''b'' = √{{overline|2}}, then ''a''<sup>2</sup> = 2''b''<sup>2</sup>. Therefore ''a''<sup>2</sup> must be even.
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| Because the square of an odd number is odd, that in turn implies that ''a'' is even. This means that ''b'' must be odd because a/b is in lowest terms.
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| On the other hand, if ''a'' is even, then ''a''<sup>2</sup> is a multiple of 4. If ''a''<sup>2</sup> is a multiple of 4 and ''a''<sup>2</sup> = 2''b''<sup>2</sup>, then 2''b''<sup>2</sup> is a multiple of 4, and therefore ''b''<sup>2</sup> is even, and so is ''b''.
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| So ''b'' is odd and even, a contradiction. Therefore the initial assumption—that √{{overline|2}} can be expressed as a fraction—must be false.
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| ===The length of the hypotenuse ===
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| The method of proof by contradiction has also been used to show that for any [[Degeneracy (mathematics)|non-degenerate]] [[right triangle]], the length of the hypotenuse is less than the sum of the lengths of the two remaining sides.<ref>{{cite web|url=http://www.cs.utexas.edu/~pstone/Courses/313Hfall12/resources/week2a-pp4.pdf|title=Logic, Sets, and Functions: Honors|last=Stone|first=Peter|work=Course materials|publisher=Department of Computer Sciences, The University of Texas at Austin|accessdate=6 February 2013|location=pp 14–23}}</ref> The proof relies on the [[Pythagorean theorem]]. Letting ''c'' be the length of the hypotenuse and ''a'' and ''b'' the lengths of the legs, the claim is that ''a'' + ''b'' > ''c''.
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| The claim is negated to assume that ''a'' + ''b'' ≤ ''c''. Squaring both sides results in (''a'' + ''b'')<sup>2</sup> ≤ ''c''<sup>2</sup> or, equivalently, ''a''<sup>2</sup> + 2''ab'' + ''b''<sup>2</sup> ≤ ''c''<sup>2</sup>. A triangle is non-degenerate if each edge has positive length, so it may be assumed that ''a'' and ''b'' are greater than 0. Therefore, ''a''<sup>2</sup> + ''b''<sup>2</sup> < ''a''<sup>2</sup> + 2''ab'' + ''b''<sup>2</sup> ≤ ''c''<sup>2</sup>. The [[transitive relation]] may be reduced to ''a''<sup>2</sup> + ''b''<sup>2</sup> < ''c''<sup>2</sup>. It is known from the Pythagorean theorem that ''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>. This results in a contradiction since strict inequality and equality are [[Mutually exclusive events|mutually exclusive]]. The latter was a result of the Pythagorean theorem and the former the assumption that ''a'' + ''b'' ≤ ''c''. The contradiction means that it is impossible for both to be true and it is known that the Pythagorean theorem holds. It follows that the assumption that ''a'' + ''b'' ≤ ''c'' must be false and hence ''a'' + ''b'' > ''c'', proving the claim.
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| ===No least positive rational number===
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| <!---redundant, compared with the lead---distinction between proving p and ¬p doesn't matter in classical logic; rule for proving p isn't accepted in intuitionistic logic---
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| Say we wish to disprove proposition ''p''. The procedure is to show that assuming ''p'' leads to a logical contradiction. Thus, according to the law of non-contradiction, ''p'' must be false.
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| Say instead we wish to prove proposition ''p''. We can proceed by assuming "not ''p''" (i.e. that ''p'' is false), and show that it leads to a logical contradiction. Thus, according to the law of non-contradiction, "not ''p''" must be false, and so, according to the [[law of the excluded middle]], ''p'' is true.
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| In symbols:
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| To disprove ''p'': one uses the [[tautology (logic)|tautology]] (''p'' → (''R'' ∧ ¬''R'')) → ¬''p'', where ''R'' is any proposition and the ∧ symbol is taken to mean "and". Assuming ''p'', one proves ''R'' and ''¬R'', and concludes from this that ''p'' → (''R'' ∧ ¬''R''). This and the tautology together imply ''¬p''.
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| To prove ''p'': one uses the tautology (¬''p'' → (''R'' ∧ ¬''R'')) → ''p'' where ''R'' is any proposition. Assuming ¬''p'', one proves ''R'' and ¬''R'', and concludes from this that ¬''p'' → (''R'' ∧ ¬''R''). This and the tautology together imply ''p''.
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| For a simple example of the first kind,
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| --->
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| Consider the proposition, ''P'': "there is no smallest rational number greater than 0". In a proof by contradiction, we start by assuming the opposite, ¬''P'': that there ''is'' a smallest rational number, say, ''r''.
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| Now ''r''/2 is a rational number greater than 0 and smaller than ''r''.
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| (In the above symbolic argument, "''r''/2 is the smallest rational number" would be ''Q'' and "''r'' (which is different from ''r''/2) is the smallest rational number" would be ¬''Q''.)
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| But that contradicts our initial assumption, ¬''P'', that ''r'' was the ''smallest'' rational number. So we can conclude that the original proposition, ''P'', must be true — "there is no smallest rational number greater than 0".
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| <!---redundant---
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| [Note: the choice of which statement is ''R'' and which is ¬''R'' is arbitrary.]
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| --->
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| <!---doubtful distinction between proving p and ¬p---
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| It is common to use this first type of argument with propositions such as the one above, concerning the ''non''-existence of some mathematical object. One assumes that such an object exists, and then proves that this would lead to a contradiction; thus, such an object does not exist.
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| --->
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| ===Other===
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| For other examples, see [[Square root of 2#Proofs of irrationality|proof that the square root of 2 is not rational]] (where indirect proofs different from the [[#Irrationality of the square root of 2|above]] one can be found) and [[Cantor's diagonal argument]].
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| <!---last paragraph moved down--->
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| ==In mathematical logic==<!---this doesn't belong to the example section--->
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| In [[mathematical logic]], the proof by contradiction is represented as:
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| : If
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| ::<math>S \cup \{ P \} \vdash \mathbb{F}</math>
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| : then
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| ::<math>S \vdash \neg P.</math>
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| or
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| : If
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| ::<math>S \cup \{ \neg P \} \vdash \mathbb{F}</math>
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| : then
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| ::<math>S \vdash P.</math>
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| In the above, ''P'' is the proposition we wish to disprove respectively prove; and ''S'' is a set of statements, which are the [[premise]]s—these could be, for example, the [[axiom]]s of the theory we are working in, or earlier [[theorem]]s we can build upon. We consider ''P'', or the negation of ''P'', in addition to ''S''; if this leads to a logical contradiction ''F'', then we can conclude that the statements in ''S'' lead to the negation of ''P'', or ''P'' itself, respectively.
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| Note that the [[union (set theory)|set-theoretic union]], in some contexts closely related to [[logical disjunction]] (or), is used here for sets of statements in such a way that it is more related to [[logical conjunction]] (and).
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| <!---last paragraph of former section "In mathematics" moved to here:--->
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| <!---doubtful distinction between proving p and ¬p---
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| On the other hand, it is also common to use arguments of the second type concerning the ''existence'' of some mathematical object.
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| --->
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| A particular kind of indirect proof assumes that some object doesn't exist, and then proves that this would lead to a contradiction; thus, such an object must exist. Although it is quite freely used in mathematical proofs, not every [[philosophy of mathematics|school of mathematical thought]] accepts this kind of argument as universally valid. See further [[Nonconstructive proof]].
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| ==Notation==
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| <!-- This section is linked from [[Hand of Eris]]. -->
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| Proofs by contradiction sometimes end with the word "Contradiction!". [[Isaac Barrow]] and Baermann used the notation Q.E.A., for "''quod est absurdum''" ("which is absurd"), along the lines of [[Q.E.D.]], but this notation is rarely used today.<ref>[http://robin.hartshorne.net/QED.html Hartshorne on QED and related]</ref> A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley.<ref>B. Davey and H.A. Priestley, Introduction to lattices and order, Cambridge University Press, 2002.</ref> Others sometimes used include a pair of [[Hand of Eris|opposing arrows]] (as <math>\rightarrow\!\leftarrow</math> or <math>\Rightarrow\!\Leftarrow</math>), struck-out arrows (<math>\nleftrightarrow</math>), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※).<ref>The Comprehensive LaTeX Symbol List, pg. 20. http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf</ref><ref>Gary Hardegree, ''Introduction to Modal Logic'', Chapter 2, pg. II–2. http://people.umass.edu/gmhwww/511/pdf/c02.pdf</ref> The "up tack" symbol (U+22A5: ⊥) used by philosophers and logicians (see contradiction) also appears, but is often avoided due to its usage for [[orthogonality]].
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| ==See also==
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| *[[Proof by contrapositive]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| {{wikibooks|1=Mathematical Proof|2=Methods of Proof/Proof by Contradiction|3=Proof by Contradiction}}
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| *{{cite book|last=Franklin|first=James|title=Proof in Mathematics: An Introduction|year=2011|publisher=Kew|location=chapter 6|isbn=978-0-646-54509-7|url=http://www.maths.unsw.edu.au/~jim/proofs.html}}
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| ==External links==
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| *[http://zimmer.csufresno.edu/~larryc/proofs/proofs.contradict.html Proof by Contradiction] from Larry W. Cusick's [http://zimmer.csufresno.edu/~larryc/proofs/proofs.html How To Write Proofs]
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| {{DEFAULTSORT:Proof By Contradiction}}
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| [[Category:Mathematical proofs]]
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| [[Category:Methods of proof]]
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| [[Category:Theorems in propositional logic]]
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The author's title is Zachery Dismukes and he considers it sounds quite good. Providing databases is his profession. Georgia happens to be his living area. Itis not a frequent thing but what I prefer doing does ceramics but I haven't created a dime
Feel free to surf to my web page :: http://prolexins.net/