Rational choice theory: Difference between revisions
en>Paradoctor tag cleanup |
|||
| Line 1: | Line 1: | ||
{{See also|Table of logic symbols}} | |||
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]]''. | |||
[[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> | |||
== | == Examples == | ||
===Irrationality of the square root of 2=== | |||
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. | |||
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. | |||
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''. | |||
So ''b'' is odd and even, a contradiction. Therefore the initial assumption—that √{{overline|2}} can be expressed as a fraction—must be false. | |||
===The length of the hypotenuse === | |||
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''. | |||
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. | |||
===No least positive rational number=== | |||
<!---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--- | |||
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. | |||
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. | |||
In symbols: | |||
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''. | |||
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''. | |||
For a simple example of the first kind, | |||
---> | |||
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''. | |||
Now ''r''/2 is a rational number greater than 0 and smaller than ''r''. | |||
(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''.) | |||
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". | |||
<!---redundant--- | |||
[Note: the choice of which statement is ''R'' and which is ¬''R'' is arbitrary.] | |||
---> | |||
<!---doubtful distinction between proving p and ¬p--- | |||
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. | |||
---> | |||
===Other=== | |||
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]]. | |||
<!---last paragraph moved down---> | |||
==In mathematical logic==<!---this doesn't belong to the example section---> | |||
In [[mathematical logic]], the proof by contradiction is represented as: | |||
: If | |||
::<math>S \cup \{ P \} \vdash \mathbb{F}</math> | |||
: then | |||
::<math>S \vdash \neg P.</math> | |||
or | |||
: If | |||
::<math>S \cup \{ \neg P \} \vdash \mathbb{F}</math> | |||
: then | |||
::<math>S \vdash P.</math> | |||
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. | |||
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). | |||
<!---last paragraph of former section "In mathematics" moved to here:---> | |||
<!---doubtful distinction between proving p and ¬p--- | |||
On the other hand, it is also common to use arguments of the second type concerning the ''existence'' of some mathematical object. | |||
---> | |||
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]]. | |||
==Notation== | |||
<!-- This section is linked from [[Hand of Eris]]. --> | |||
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]]. | |||
==See also== | |||
*[[Proof by contrapositive]] | |||
==References== | |||
{{Reflist}} | |||
==Further reading== | |||
{{wikibooks|1=Mathematical Proof|2=Methods of Proof/Proof by Contradiction|3=Proof by Contradiction}} | |||
*{{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}} | |||
==External links== | |||
*[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] | |||
{{DEFAULTSORT:Proof By Contradiction}} | |||
[[Category:Mathematical proofs]] | |||
[[Category:Methods of proof]] | |||
[[Category:Theorems in propositional logic]] | |||
Revision as of 07:10, 2 February 2014
DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.
We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.
The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.
Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.
is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease
In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value.
In logic, proof by contradiction is a form of proof that establishes the 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.
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."[1]
Examples
Irrationality of the square root of 2
A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational.[2] If it were 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. But if a/b = √Template:Overline, then a2 = 2b2. Therefore a2 must be even. 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.
On the other hand, if a is even, then a2 is a multiple of 4. If a2 is a multiple of 4 and a2 = 2b2, then 2b2 is a multiple of 4, and therefore b2 is even, and so is b.
So b is odd and even, a contradiction. Therefore the initial assumption—that √Template:Overline can be expressed as a fraction—must be false.
The length of the hypotenuse
The method of proof by contradiction has also been used to show that for any non-degenerate right triangle, the length of the hypotenuse is less than the sum of the lengths of the two remaining sides.[3] 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.
The claim is negated to assume that a + b ≤ c. Squaring both sides results in (a + b)2 ≤ c2 or, equivalently, a2 + 2ab + b2 ≤ c2. 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, a2 + b2 < a2 + 2ab + b2 ≤ c2. The transitive relation may be reduced to a2 + b2 < c2. It is known from the Pythagorean theorem that a2 + b2 = c2. This results in a contradiction since strict inequality and equality are 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.
No least positive rational number
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.
Now r/2 is a rational number greater than 0 and smaller than r. (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.) 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".
Other
For other examples, see proof that the square root of 2 is not rational (where indirect proofs different from the above one can be found) and Cantor's diagonal argument.
In mathematical logic
In mathematical logic, the proof by contradiction is represented as:
or
In the above, P is the proposition we wish to disprove respectively prove; and S is a set of statements, which are the premises—these could be, for example, the axioms of the theory we are working in, or earlier theorems 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.
Note that the 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).
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 school of mathematical thought accepts this kind of argument as universally valid. See further Nonconstructive proof.
Notation
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.[4] A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley.[5] Others sometimes used include a pair of opposing arrows (as or ), struck-out arrows (), a stylized form of hash (such as U+2A33: ⨳), or the "reference mark" (U+203B: ※).[6][7] 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.
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Further reading
DTZ's auction group in Singapore auctions all types of residential, workplace and retail properties, retailers, homes, accommodations, boarding houses, industrial buildings and development websites. Auctions are at the moment held as soon as a month.
Whitehaven @ Pasir Panjang – A boutique improvement nicely nestled peacefully in serene Pasir Panjang personal estate presenting a hundred and twenty rare freehold private apartments tastefully designed by the famend Ong & Ong Architect. Only a short drive away from Science Park and NUS Campus, Jade Residences, a recent Freehold condominium which offers high quality lifestyle with wonderful facilities and conveniences proper at its door steps. Its fashionable linear architectural fashion promotes peace and tranquility living nestled within the D19 personal housing enclave. Rising workplace sector leads real estate market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline International Market Perspectives - 1st Quarter 2014
There are a lot of websites out there stating to be one of the best seek for propertycondominiumhouse, and likewise some ways to discover a low cost propertycondominiumhouse. Owning a propertycondominiumhouse in Singapore is the dream of virtually all individuals in Singapore, It is likely one of the large choice we make in a lifetime. Even if you happen to're new to Property listing singapore funding, we are right here that will help you in making the best resolution to purchase a propertycondominiumhouse at the least expensive value.
Jun 18 ROCHESTER in MIXED USE IMPROVEMENT $1338000 / 1br - 861ft² - (THE ROCHESTER CLOSE TO NORTH BUONA VISTA RD) pic real property - by broker Jun 18 MIXED USE IMPROVEMENT @ ROCHESTER @ ROCHESTER PK $1880000 / 1br - 1281ft² - (ROCHESTER CLOSE TO NORTH BUONA VISTA) pic real estate - by broker Tue 17 Jun Jun 17 Sunny Artwork Deco Gem Near Seashore-Super Deal!!! $103600 / 2br - 980ft² - (Ventnor) pic actual estate - by owner Jun 17 Freehold semi-d for rent (Jalan Rebana ) $7000000 / 5909ft² - (Jalan Rebana ) actual property - by dealer Jun sixteen Ascent @ 456 in D12 (456 Balestier Highway,Singapore) pic real property - by proprietor Jun 16 RETAIL SHOP AT SIM LIM SQUARE FOR SALE, IT MALL, ROCHOR, BUGIS MRT $2000000 / 506ft² - (ROCHOR, BUGIS MRT) pic real estate - by dealer HDB Scheme Any DBSS BTO
In case you are eligible to purchase landed houses (open solely to Singapore residents) it is without doubt one of the best property investment choices. Landed housing varieties solely a small fraction of available residential property in Singapore, due to shortage of land right here. In the long term it should hold its worth and appreciate as the supply is small. In truth, landed housing costs have risen the most, having doubled within the last eight years or so. However he got here back the following day with two suitcases full of money. Typically we've got to clarify to such folks that there are rules and paperwork in Singapore and you can't just buy a home like that,' she said. For conveyancing matters there shall be a recommendedLondon Regulation agency familiar with Singapore London propertyinvestors to symbolize you
Sales transaction volumes have been expected to hit four,000 units for 2012, close to the mixed EC gross sales volume in 2010 and 2011, in accordance with Savills Singapore. Nevertheless the last quarter was weak. In Q4 2012, sales transactions were 22.8% down q-q to 7,931 units, in line with the URA. The quarterly sales discount was felt throughout the board. When the sale just starts, I am not in a hurry to buy. It's completely different from a private sale open for privileged clients for one day solely. Orchard / Holland (D09-10) House For Sale The Tembusu is a singular large freehold land outdoors the central area. Designed by multiple award-profitable architects Arc Studio Architecture + Urbanism, the event is targeted for launch in mid 2013. Post your Property Condos Close to MRT
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- Proof by Contradiction from Larry W. Cusick's How To Write Proofs
- ↑ G. H. Hardy, A Mathematician's Apology; Cambridge University Press, 1992. ISBN 9780521427067. p. 94.
- ↑ Template:Cite web
- ↑ Template:Cite web
- ↑ Hartshorne on QED and related
- ↑ B. Davey and H.A. Priestley, Introduction to lattices and order, Cambridge University Press, 2002.
- ↑ The Comprehensive LaTeX Symbol List, pg. 20. http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf
- ↑ Gary Hardegree, Introduction to Modal Logic, Chapter 2, pg. II–2. http://people.umass.edu/gmhwww/511/pdf/c02.pdf