
	Where results make sense

Topic: Direct proof

Related Topics

proof theory

burden of proof

proof by contradiction

alcoholic proof

Interactive proof system

Zero knowledge proofs

Conditional proof

Invalid proof

structural proof theory

consistency proof

proof by exhaustion

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	Mathematical proof
		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.ebroadcast.com.au /lookup/encyclopedia/pr/Proof_(math).html (399 words)

	CATHOLIC ENCYCLOPEDIA: Proof
		Proof is the establishment of a disputed or controverted matter by lawful means or arguments.
		Proof is perfect, or complete, when it produces full conviction, and enables the judge without further investigation to pronounce sentence: imperfect, or incomplete, if it begets probability only.
		Two imperfect proofs cannot constitute perfect proof in criminal cases, in which proof must be clearer than the noonday sun; in matrimonial cases, when there is question of the validity of a marriage already contracted; or in civil actions of a grave character.
	www.newadvent.org /cathen/12454c.htm (1514 words)

	Proof techniques (Site not responding. Last check: 2007-10-10)
		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)

	Elementary Number Theory and Methods of Proof
		Start the proof by supposing that x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true.
		Writing proofs is similar to writing a computer program based on a set of specifications: organize your thoughts, declare your variables, document thoroughly [italicized brackets here], and follow a logical progression.
		Understanding the ideas of generalizing from the generic particular and the method of direct proof, allows one to write the beginnings of a proof even for a theorem not well understood.
	people.uncw.edu /tompkinsj/133/proofs/proo.htm (1437 words)

	Methods of mathematics proof
		A direct proof with many steps is like crossing a stream by stepping on steppable protuberances in the water.
		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.
		Since both RAA and proof by contrapositive are indirect proofs, it is clearer to the reader of the proof not to mention RAA as just an indirect proof.
	www.mathpath.org /proof/proof.methods.htm (2459 words)

	The Use of Logic in Teaching Proof
		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.
		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.
		Once students have a sense for the overall structure of a proof, which is largely determined by the logical form of the statement to be proved and of the definitions of its terms, I encourage them to try to identify the crux, or essential idea of the proof.
	condor.depaul.edu /~sepp/MAT140/TeachingProof.htm (3749 words)

	MACM 101 D1
		A direct proof of the contrapositive form of an implication theorem.
		Specifically, the conclusion is assumed false and a contradiction is derived using the direct proof technique.
		Remarks: An indirect proof (proof by contradiction) is often regarded as a special case of proof by contrapositive.
	www.cs.sfu.ca /fas-info/cs/CC/101.MACM/stevenp/Proof.html (1416 words)

	methods of proof
		Direct proof is a technique where we prove a conditional by showing that the conclusion B must be true when the premise A is true.
		The method of proof by contradiction is based on the fact that either a statement to be proven is true or it is false, but never both.
		Another method of proof is called the choose method, which is based on the following idea: To show that every element in a set satisfies a particular property, suppose an element x is a particular but arbitrarily chosen element of the set, and show that x satisfies that property.
	cse.stanford.edu /classes/cs103a/h24RProofs.htm (4876 words)

	[No title]
		I often find that it is easiest to start with a proof by contradiction and then develop a direct proof.
		Make sure that you write the correct assumption to contradict if you are doing a proof by contradiction or the correct contrapositive if you are using this method—sometimes this is the most difficult part of the proof.
		Structure of Mathematical Arguments: Proof by implication: A implies B (direct proof) Use accepted axioms, assumptions that you have already chosen to make, and theorems which have already been proved to get from A to B. Example (from Simon and Blume p.
	ucsu.colorado.edu /~lipscomm/Proofs.doc (1208 words)

	Mathematical proof - Wikipedia, the free encyclopedia
		In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true.
		That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics.
		Purely formal proofs are considered in proof theory.
	en.wikipedia.org /wiki/Mathematical_proof (1331 words)

	SparkNotes: Geometric Proofs: The Structure of a Proof
		A two-column geometric proof consists of a list of statements, and the reasons that we know those statements are true.
		Every step of the proof (that is, every conclusion that is made) is a row in the two-column proof.
		This is the step of the proof in which you actually find out how the proof is to be made, and whether or not you are able to prove what is asked.
	www.sparknotes.com /math/geometry3/geometricproofs/section1.html (394 words)

	Synopses of Topics - Direct Proofs (Site not responding. Last check: 2007-10-10)
		There are five basic strategies to prove a statement of the form P ==> Q: 1) Direct proof; 2) Direct proof of the contrapositive; 3) Proof by contradiction; 4) Mathematical induction; 5) Case by case analysis.
		A direct proof consists of an initial statement connected to a statement to be proven by a finite number of intermediate statements, P
		Once a direct proof of P ==> Q is constructed, the logic of it is completely transparent.
	math.usask.ca /emr/dirp.html (615 words)

	Illustration of proper mathematics proof and solution
		The proof of the second assertion rests on the observation that a convex n-gon can be partitioned into n-2 triangles and that the sum of the angles from all these triangles equals the sum of the internal angles of the polygon.
		To complete the proof that the maximum number of acute angles is three, we need to prove that, for any n, there is a convex n-gon with three acute angles.
		Aside from being motivators of proof design, illustrations serve two uses: (a) in clarifying an argument in a proof and (b) in showing that a statement is false.
	www.mathpath.org /MathPath_Quiz_2003_files/example.htm (2260 words)

	Direct Proofs
		A direct poof should be thought of as a flow of implications beginning with "P" and ending with "Q".
		Most proofs are (and should be) direct proofs.
		Always try direct proof first, unless you have a good reason not to.
	zimmer.csufresno.edu /~larryc/proofs/proofs.direct.html (667 words)

	SparkNotes: Geometric Proofs: Terms
		Direct Proof - A proof in which the conclusion is drawn directly from previous conclusions, starting with the first statement.
		There are two major types of proofs: direct proofs and indirect proofs.
		Indirect Proof - A proof in which a statement is shown to be true because the assumption that its negation is true leads to a contradiction.
	www.sparknotes.com /math/geometry3/geometricproofs/terms.html (244 words)

	Making Mathematics: Mathematics Tools: Proof by Contradiction (Site not responding. Last check: 2007-10-10)
		For example, to show that the square root of two is irrational, we cannot directly test and reject the infinite number of rational numbers whose square might be two.
		One way to understand it is to note that you are creating a direct proof of the contrapositive of your original statement (you are proving if not B, then not A).
		An indirect proof establishes that the opposite conclusion is not consistent with the premise and that, therefore, the original conclusion must be true.
	www2.edc.org /makingmath/mathtools/contradiction/contradiction.asp (419 words)

	Forms of Proof
		A proof written in narrative form is very often arranged in one of a few basic forms.
		What usually happens is that you think of statements which imply Q, statements which imply them (backing up), and at the same time you think of statements which P implies, statements which they imply (going forward), and so on, until your chain meets in the middle (if you are lucky).
		If a theorem has the form “If P then Q” and the proof begins with “Assume Q is false…”, that is a CLUE that the form of the proof is by contrapositive!
	www.abstractmath.org /MM/MMFormsProof.htm (919 words)

	Section1pt4
		Remember that you are trying to convince your reader of the validity of your proof through the clarity and simplicity of your organization and logical reasoning.
		Finish your proof with a clear statement of that which was to be proven.
		Remember that you are trying to convince your reader of the validity of the proof through the clarity and simplicity of your organization and logical reasoning.
	www.math.ucdavis.edu /~kouba/TheProofPage/Section1pt4/Section1pt4.html (742 words)

	Strategies for Proofs
		It is important to remember that there are many different proof techniques, and in general there is no way of knowing in advance which technique will be most effective.
		The proof that the square root of 2 is irrational runs along those lines.
		Using a few test cases may help gain understanding and provide a clue on how to solve a proof, but is not a proof itself.
	pages.prodigy.net /bhare/si_prftech.html (1353 words)

	context :: direct proof of dark matter
		Dark matter and normal matter have been wrenched apart by the tremendous collision of two large clusters of galaxies.
		In galaxy clusters, the normal matter, like the atoms that make up the stars, planets, and everything on Earth, is primarily in the form of hot gas and stars.
		most direct measurement of dark matter allows study of its nature.
	straddle3.net /context/03/en/2006_09_01.html (768 words)

	Section 2.1 Direct Proof and Counterexample
		Another way is to give a set of directions for finding such an x.
		The disadvantage of a nonconstructive proof is that it may give virtually no clue about where or how x may be found.
		Start the proof by assuming x is an element of M for which the hypothesis P(x) is true.
	users.csc.tntech.edu /~srini/DM/chapters/review2.1.html (497 words)

	[No title]
		In a direct proof, you assume that P is true, then use inference rules and other facts to prove that Q is true.
		In these cases, you can perhaps "apply a logical equivalence" to the formula you are trying to prove, then prove this equivalent formula.
		an indirect proof of P --> Q is: ~Q --> ~P so, assume Q is false, and show that then P is false.
	www.cs.wustl.edu /~pless/cs201/lecture06directProofs.htm (599 words)

	Direct Proofs (Site not responding. Last check: 2007-10-10)
		So the difficulty in a direct proof is finding these connections between certain facts.
		One might compare a direct proof to crossing a river using stepping stones.
		In first attempts at proofs, students often assume q and derive a true statement such as ``1 = 1'' and conclude that q is true.
	www.math.csusb.edu /notes/proofs/pfnot/node5.html (539 words)

	XANTE CL30 Printer Features and Benefits / Print Direct / Proof / Job Storage / Spooling / Secure
		XANTÉ Print Direct allows you to send a selected PDF file directly to the printer, a faster and easier process than using Adobe Acrobat separately before printing.
		If you need to load a specific paper for the print job this feature will allow you to do so before the job is released to print.
		Proof and print allows printing of a single copy of a document for checking before printing multiple copies of the same document.
	www.xante.com /features_benefits/features_benefits_clx.aspx (671 words)

	Dvorak Uncensored » NASA Establishes Direct Proof of Dark Matter
		Further proof of how little we always seem to know in relation to what the future of discoveries holds for humanity.
		Proof in this case I’m sure is a media invention.
		As was stated earlier the only thing this kind of discovery can hope to achieve is to be classified as strong evidence.
	www.dvorak.org /blog/?p=6713 (1004 words)

	Chandra :: Photo Album :: 1E 0657-56 :: 21 Aug 06
		The concentration of mass is determined using the effect of so-called gravitational lensing, where light from the distant objects is distorted by intervening matter.
		Most of the matter in the clusters (blue) is clearly separate from the normal matter (pink), giving direct evidence that nearly all of the matter in the clusters is dark.
		The hot gas in each cluster was slowed by a drag force, similar to air resistance, during the collision.
	chandra.harvard.edu /photo/2006/1e0657 (402 words)

	MAT 140: Discrete Mathematics I
		Write the word “Proof” at the beginning of a proof.
		That is, suppose...,” and continue this sentence by carefully writing the negation of the statement to be proved.
		A proof by mathematical induction consists of two parts.
	condor.depaul.edu /~sepp/MAT140/ProofTipsW04.htm (336 words)

	CERN Courier - US team finds direct proof f - IOP Publishing - article
		The idea of dark matter in the universe dates back to the 1930s, with the observation that the gravitational force on the visible matter in clusters of galaxies could not fully account for their behaviour, implying some alteration to gravity, or the existence of non-luminous, invisible matter.
		The new study, however, has discovered a system in which the inferred dark matter is not coincident with the observable matter, and the difference in position is too great to be accounted for by modifying gravity.
		The team from the universities of Arizona and Florida, the Kavli Institute for Particle Astrophysics and Cosmology, and the Harvard-Smithsonian-Center for Astrophysics has combined observations from various telescopes to build a picture of what is happening in the galaxy cluster 1E0657-558.
	cerncourier.com /main/article/46/8/7 (397 words)

	Special Topic: Proof Strategies (Site not responding. Last check: 2007-10-10)
		That is, they define a formal system which we can then use to study inference, or meaning or any other strange we may be able to model use the syntactic tools of the system.
		Constructing proofs has its own set of rules.
		There are many short, pithy proofs that may violate these rules, but those are for the grand masters of proof to find.
	www.personal.kent.edu /~pbohanbr/Webpage/formal2/node7.html (102 words)

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