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| {{See also|Table of logic symbols}}
| | Nice to meet you, I am Marvella Shryock. He used to be unemployed but now he is a pc operator but his marketing by no means comes. To perform baseball is the pastime he will by no means quit performing. North Dakota is her birth location but she will have to move 1 working day or an additional.<br><br>Also visit my web site: at home std test ([http://Www.buzzbit.net/user/AFilson redirected here]) |
| 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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Nice to meet you, I am Marvella Shryock. He used to be unemployed but now he is a pc operator but his marketing by no means comes. To perform baseball is the pastime he will by no means quit performing. North Dakota is her birth location but she will have to move 1 working day or an additional.
Also visit my web site: at home std test (redirected here)