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Topic: Proof by contradiction

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proof theory

burden of proof

proof by contradiction

alcoholic proof

Interactive proof system

Zero knowledge proofs

Conditional proof

Invalid proof

structural proof theory

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	Contradiction - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-06)
		The idea of a contradiction as a conflict based in the structure of a social situation is not unique to Marxian thought.
		For example, for liberal thinkers, the problem of public goods may be interpreted as a "contradiction": there is a conflict between what's good for society (the production of a public good such as national defense) and what's good for individual free riders, who refuse to pay the costs of the public good.
		To construct a proof by contradiction, then, you construct a valid proof from a set of premises to a conclusion that is a logical contradiction.
	encyclopedia.worldsearch.com /contradiction.htm (1120 words)

	Learn more about Mathematical proof in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
		Proof by contradiction: where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false.
		A probabilistic proof should mean a proof in which an example is shown to exist by methods of probability theory - not an argument that a theorem is 'probably' true.
	www.onlineencyclopedia.org /m/ma/mathematical_proof.html (559 words)

	Proofs by Contradiction
		Proof by contradiction is often used when you wish to prove the impossibility of something.
		(Proof by Contradiction.) Assume to the contrary there is a rational number p/q, in reduced form, with p not equal to zero, that satisfies the equation.
		Proof by Contradiction is often the most natural way to prove the converse of an already proved theorem.
	zimmer.csufresno.edu /~larryc/proofs/proofs.contradict.html (832 words)

	Reductio ad absurdum - Wikipedia, the free encyclopedia
		For examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.
		Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of argument as universally valid.
		It is important to note that to form a valid proof, it must be demonstrated that the assumption being made for the sake of argument implies a property that is actually false in the mathematical system being used.
	en.wikipedia.org /wiki/Proof_by_contradiction (1112 words)

	Proof techniques
		A proof is a sequence of statements, each of which is an axiom, previously proved theorem, or is derived from previous statements in the sequence by means of a rule of inference.
		A direct proof of a statement of the form A→B, begins with the assumptions encapsulated in A, and proceeds to construct a sequences of statements each of which is an axiom, previously proved theorem, or follows from previous statements by a rule of inference.
		Proof by contradiction is also known as indirect proof, top-down proof, or goal directed proof.
	cs.wwc.edu /KU/Logic/ProofTech.html (400 words)

	Proof By Contradiction - Science Forums and Debate (Site not responding. Last check: 2007-11-06)
		I've come to see that contradictions are often used for quite short proofs most of the time, and they provide a quick and easy way to prove statements that are not as approachable by a direct method.
		This isn't the case for all contradictive proofs though, as obviously quite a few are long winded.
		I suggest when you look through a proof that uses a contradiction, look at the way the statement goes through one logical step to another and see how the statement is being manipulated.
	www.scienceforums.net /forums/showthread.php?p=47278 (378 words)

	Methods of mathematics proof
		We recommend that a Proof by Contradiction be one that begins with p and ~q and ends up obtaining the negation of the premise, and that a Reductio Ad Absurdum Proof be one that ends up obtaining any contradiction of a known truth.
		While it is not incorrect to call an RAA proof as an indirect proof, if you insist on mentioning 'indirect', it might be better to say 'indirect proof by contradiction' or 'indirect proof by contrapositive' as the case may be, or just 'proof by contradiction' or 'proof by contrapositive.'
		Although RAA proofs are often easier and more convenient, a direct proof is preferred for the reason that RAA depends for its validity on the assumption that the unprovability of the negation of p is tantamount to the provability of the negation of the negation of p.
	www.mathpath.org /proof/proof.methods.htm (2455 words)

	New Page 1
		While an exhaustive proof is completely legitimate, it is worth pointing out that it is actually a relatively uncommon proof technique since it is rare that the number of possibilities is finite enough for us to test every case.
		In formal direct proofs we assume that the hypothesis are true and attempt to deduce the conclusion through a series of known and accepted transformations (inference and equivalence rules in these sections).
		Although it is not completely clear from the above example, proof by contradiction is most often used when we are arguing that Q is not true since the negation step turns it to a true statement.
	www.cs.uni.edu /~schafer/courses/080/sessions/s28.htm (1019 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Later, after introducing proof by contraposition and proof by contradiction as well as direct proof, one can help students keep the three basic proof methods separate by pointing out that while for each method there is something supposed and something to be shown, these “somethings'“ are dramatically different in each case.
		In a proof by contradiction one supposes that the entire statement to be proved is false, and one shows that this supposition leads to a contradiction.
		But for most of the proofs undergraduate students are asked to construct, the majority of this task is achieved through a logico-linguistic analysis of definitions because the inner structure of a straightforward, or routine, mathematical proof is largely determined by the meanings of the terms in the hypothesis and the conclusion.
	condor.depaul.edu /~sepp/PREP/TeachingProof.doc (3863 words)

	Teaching Math: Grades 9-12: Reasoning and Proof
		In this type of proof, the logical negation of a conjecture is taken as the premise.
		Students should become comfortable with this type of proof, which builds in part on their earlier understanding that counterexamples can be used to show that a general conjecture does not hold.
		Proof by contradiction can be used to establish the validity of a conjecture.
	www.learner.org /channel/courses/teachingmath/grades9_12/session_04/section_03_c.html (316 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Zeno's Paradox of Plurality and Proof by Contradiction
		Proof by contradiction, or indirect proof, presents notorious difficulties from a pedagogical point of view, and many approaches to make it more accessible to students have been proposed and explored.
		This is a contradiction, therefore, given the truth of our lemma (0), our supposition that the square root of 2 equals a/b follows the way of seeming, and not the way of truth.
	cs.wwc.edu /~aabyan/CII/TEMP/Zeno2.html (4021 words)

	Maggie Johnson
		Formal Proof: Every step in the proof is provided (i.e., no steps are left out), a fixed set of rules are used as explanations of intermediate conclusions; usually presented in a highly stylized, formal way.
		Informal Proof: Usually stated in English, in paragraph form; less formal and the more obvious steps are left out.
		Contradiction: any claim that cannot be true, or any set of claims that cannot all be true in any single situation.
	cse.stanford.edu /classes/cs103a/h11Proofs2.htm (288 words)

	Cantor's diagonal argument (Site not responding. Last check: 2007-11-06)
		Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite.
		Cantor's original proof shows that the interval [0,1] is not countably infinite.
		For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument.
	www.sciencedaily.com /encyclopedia/cantor_s_diagonal_argument (860 words)

	[Dpf] Totalities Including Contradictions and Proof by Contradiction (Site not responding. Last check: 2007-11-06)
		The method of proof by contradiction, also called Reductio ad absurdum (literally ‘reduction to absurdity’), begins a logical proof with a provisional assumption that is a negation of a statement.
		Applying proof by contradiction to a world whose totality involves the existence of contradictions may render the proof’s results invalid.
		The truth of this seemingly absurd contradiction “’A’ and not ‘A’,” may be the open space or the looseness in the system that allows for emergence to enter.
	lists.econ.utah.edu /pipermail/dpf/2004-December/000100.html (367 words)

	Contradiction Command Help (Site not responding. Last check: 2007-11-06)
		Proof Designer will create a subproof in which it is assumed that the goal is false, and then it will say that you must prove a contradiction to complete the subproof.
		Proof Designer will indicate that you have to prove the negation of the selected given in order to arrive at the desired contradiction.
		Proof Designer will create a subproof in which it is assumed that the goal is false, and then it will say that you must prove the negation of the selected given to arrive at a contradiction.
	www.cs.amherst.edu /~djv/pd/help/Contrad.html (212 words)

	Irrational numbers and proof by contradiction
		In the book, I note that a number of ‘conditions’ which economists impose upon their models are actually ‘proofs by contradiction’ that their theories contain errors.
		In the book, I promised to give an example of how mathematicians used ‘proof by contradiction’ properly to transcend an old belief that all numbers were rational, and ushered in one of many great leaps in our understanding of mathematical logic.
		We have a contradiction, and therefore have proven that the square root of two can’t be expressed as the ratio of two integers.
	www.debunking-economics.com /Maths/Irrational.htm (549 words)

	Dangers of proof by contradiction (Site not responding. Last check: 2007-11-06)
		Proof by contradiction is a powerful mathematical technique: if you want to prove X, start by assuming X is false and then derive consequences.
		If you reach a contradiction with something you know is true, then the only possible problem can be in your initial assumption that X is false.
		Then it looks like the proof is done, but unfortunately the contradiction has nothing to do with the initial assumption, and comes solely from the mistake in the middle.
	research.microsoft.com /~cohn/Thoughts/contradiction.html (407 words)

	[No title]
		A proof of a thesis under these labels is invoked whenever the thesis cannot be proved constructively by deriving what it is supposed to prove.
		if so, then, by arguing that contradiction of the contradiction of a thesis is equivalent to the thesis (double negation is positive, just as double clicking of light switch restores the state), the claim is made that the thesis has been proven.
		In factor language, "proof by contradicction" (double negation) means that a number is equivalent to the complement of its complement.
	members.fortunecity.com /jonhays/proofby.htm (718 words)

	Project 1 (Site not responding. Last check: 2007-11-06)
		A contradiction is a statement form that is always false.
		Proof by contradiction is a widely used method of mathematical proof.
		Necessary details have been omitted throughout the proof and replaced with the symbol (???).Read the proof, making sure you understand the reasoning and how the method of proof by contradiction is used.
	math.colorado.edu /~amyc/project1/proj1.html (525 words)

	Proof Strategies
		Usually, when you are working on a proof, you should use the logical forms of the givens and goals to guide you in choosing what proof strategies to use.
		Use proof by contradiction; i.e., assume P is true and try to reach a contradiction.
		Proof by contradiction: Assume the goal is false and derive a contradiction.
	www.cs.amherst.edu /~djv/pd/help/Strategies.html (1263 words)

	Teaching Math: Grades 9-12: Reasoning and Proof
		Suppose one of your students thinks that demonstrating that a mathematical conjecture is true for 1,000 instances constitutes formal proof.
		Put yourself in the position of a student who has always been very proficient at mathematical procedures, receiving consistently high marks and expecting to be one of the fastest students to figure something out or hand something in.
		Now consider that this student has encountered a formal proof of a geometry conjecture for the first time and is baffled by it.
	www.learner.org /channel/courses/teachingmath/grades9_12/session_04/section_03_g.html (659 words)

	Sets
		are shown to imply the conclusion C. In indirect proofs, we have (1) proof of the contrapositive, (2) proof by contradiction, and (3) proof by cases.
		When we use proof by contradiction, we assume the negation of the conclusion and use it together with the set of hypotheses to obtain a contradiction or an absurdity:
		A vacuous proof is a proof of an implication
	cse.unl.edu /~lksoh/Classes/CSCE235_Spring02/sup4.html (690 words)

	[No title]
		Suspending proof of level 4 subgoal for proof by contradiction LP0.1.96: resume by cases is_inj (fc:fun[T,U]) Creating subgoals for proof by cases Case hypotheses: inj_inf4CaseHyp.1.1: is_inj(fc) inj_inf4CaseHyp.1.2: ~is_inj(fc) Same subgoal for all cases: false Attempting to prove level 5 subgoal for case 1 (out of 2) Added hypothesis inj_inf4CaseHyp.1.1 to the system.
		Suspending proof of level 7 subgoal for proof by contradiction LP0.1.138: crit-pairs inj_inf5 with Iterate The following equations are critical pairs between rewrite rules inj_inf5.3 and Iterate.3.
		Suspending proof of level 5 subgoal for proof by contradiction LP0.1.369: declare operator wc: -> T LP0.1.370: fix x as wc in Contra A prenex-existential quantifier in formula inj_rr1ContraHyp.1 has been eliminated to produce formula inj_rr1.3, wc \in (Ac - wc) The formulas cannot be ordered using the current ordering.
	www.cs.brandeis.edu /~cokokola/vldbscripts.lp (12477 words)

	Taxonomy of proof: contrapositive (Site not responding. Last check: 2007-11-06)
		Since we are assuming that the result is not true and end up by contradicting our assumptions, this is a kind of proof by contradiction known as proof by contrapositive.
		We used it back in Section 1 for the very first proof; it is used quite a lot.
		This flexibility in proof methods is yet another reason to prefer the two-step approach to if and only if proofs to the monolithic approach.
	www-cs-students.stanford.edu /~csilvers/proof/node6.html (163 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		Date: 09/25/97 at 22:36:25 From: Doctor Pete Subject: Re: Geometric Proofs Hi, Well, it seems that you've done essentially what is a sketch of the proof.
		To write a proof, you need to be rigorous and actually provide some sort of mathematics to support your claims; for example, you say that you have determined that Sqrt[2] is not an integer, but how?
		The proof, which is a classic, is by contradiction.
	mathforum.org /library/drmath/view/54963.html (483 words)

	PHIL 2340 - chapter 5 - 6.2 (Site not responding. Last check: 2007-11-06)
		In proof by cases, we determine what possibilities there are, and then attempt to show that our conclusion follows no matter which of these possibilities obtains.
		In proof by contradiction, we assume the opposite of what we want to prove, and show that this assumption leads to a contradiction.
		The general idea: if a sentence leads to a contradiction, it must not be true (so its negation must be true).
	www.trinity.edu /cbrown/logic/chapter05.html (923 words)

	OT: Gorians: Don't get over it yet.... (Site not responding. Last check: 2007-11-06)
		Let's say that you want to prove sentence A. To do a proof by contradiction, you assume the negation of A, and use this assumption to "prove" something that everyone knows to be false.
		If none of the other premises of your proof are suspect, then it must be the negation of A that is suspect, and therefore you have proved that sentence A is true.
		Now that you understand proof by contradiction, you will no doubt be able to see why so many of your crazy assertions cannot be true.
	gaffa.org /pipermail/love-hounds/2001-March/015065.html (466 words)

	Contradiction article - Contradiction logic negation of non-contradiction "Contradiction" outside - What-Means.com (Site not responding. Last check: 2007-11-06)
		Contradiction article - Contradiction logic negation of non-contradiction "Contradiction" outside - What-Means.com
		This is one interpretation of Hegel's view of contradictions, seen for example in Paul Deising, Hegel's Dialectical Political Economy (ISBN 0813391318).
		Contradiction article - Contradiction definition - what means Contradiction
	www.what-means.com /encyclopedia/But (1059 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-11-06)
		It is true that experienced mathematicians communicate the proofs of their theorems in a sparse language that wastes no ink on the page.
		What matters the most is showing that the proof has been pursued logically, and that there are no leaps or gaps in the path to the conclusion.
		Understanding the three proof techniques of induction, deduction, and proof by contradiction can often give you ideas for approaches to take when you are struggling with a problem.
	www.cs.uidaho.edu /~casey931/mega-math/gloss/math/proof.html (296 words)

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