From Wikipedia, the free encyclopedia
Jump to:
navig=
ation,=20
sear=
chIn mathematics, a =
proof=20
is a convincing demonstration (within the accepted standards of the =
field) that=20
some mathematical=
=20
statement is necessarily true[1][2].=20
Proofs are obtained from deductive =
reasoning,=20
rather than from inductive =
or empirical=20
arguments. That is, a proof must demonstrate that a statement is true in =
all=20
cases, without a single exception. An unproved proposition that is =
believed to=20
be true is known as a conjecture.
The statement that is proved is often called a theorem[1].=20
Once a theorem is proved, it can be used as the basis to prove further=20
statements. A theorem may also be referred to as a lemma, =
especially if=20
it is intended for use as a stepping stone in the proof of another =
theorem.
Proofs employ logic but usually =
include some=20
amount of natural =
language which=20
usually admits some ambiguity. In fact, the vast majority of proofs in =
written=20
mathematics can be considered as applications of rigorous informal logic. =
Purely formal=20
proofs, written in symbolic language instead of natural language, =
are=20
considered in proof theory. The =
distinction between formal=20
and informal proofs has led to much examination of current and =
historical mathematical =
practice, qua=
si-empiricism=20
in mathematics, and so-called folk =
mathematics (in=20
both senses of that term). The philosoph=
y of=20
mathematics is concerned with the role of language and logic in =
proofs, and=20
mathemati=
cs as a=20
language.
|
Contents
- 1 History and=20
etymology=20
- 2 Nature and=20
purpose=20
- 3 Methods of =
proof=20
- 3.1 Direct =
proof=20
- 3.2 Proof by =
mathematical=20
induction=20
- 3.3 Proof by=20
transposition=20
- 3.4 Proof by=20
contradiction=20
- 3.5 Proof by=20
construction=20
- 3.6 Proof by=20
exhaustion=20
- 3.7 Probabilistic=20
proof=20
- 3.8 Combinatorial=20
proof=20
- 3.9 Nonconstructive=20
proof=20
- 3.10 Visual=20
proof=20
- 3.11 Elementary =
proof=20
- 3.12 Two-column =
proof=20
- 3.13 Statistical proofs in=20
pure mathematics=20
- 3.14 Computer-assisted=20
proofs
- 4 Undecidable=20
statements=20
- 5 Heuristic =
mathematics and=20
experimental mathematics=20
- 6 Related =
concepts=20
- 6.1 Colloquial =
use of=20
"mathematical proof"=20
- 6.2 Statistical =
proof using=20
data=20
- 6.3 Inductive =
logic proofs=20
and Bayesian analysis=20
- 6.4 Proofs as =
mental=20
objects=20
- 6.5 Influence =
of=20
mathematical proof methods outside mathematics =
- <=
SPAN=20
class=3Dtocnumber>7 Ending a =
proof=20
- 8 See =
also=20
- 9 References=20
- 10 Sources=20
- <=
SPAN=20
class=3Dtocnumber>11 External =
links=20
|
[edit]=20
History and=20
etymology
For more details on this =
topic, see
History of =
logic.Plausibility arguments using heuristic devices such as pictures and =
analogies=20
preceded strict mathematical proof.[3]=20
The development of mathematical proof is primarily the product of early =
Greek=20
civilization. Thales (624=E2=80=93546 =
BCE) proved some=20
theorems in geometry. Eudoxus =
(408=E2=80=93355 BCE) and Theaetetus=20
(417=E2=80=93369 BCE) formulated theorems but did not prove them. Aristotle =
(384=E2=80=93322 BCE) said=20
definitions should describe the concept being defined in terms of other =
concepts=20
already known. Euclid (300 BCE) began =
with undefined terms =
and axioms (propositions =
regarding=20
the undefined terms assumed to be self-evidently true, from the Greek =
=E2=80=9Caxios=E2=80=9D=20
meaning =E2=80=9Csomething worthy=E2=80=9D) and used these to prove =
theorems using deductive =
logic. Modern=20
proof=20
theory treats proofs as inductively defined data structures. There =
is no=20
longer an assumption that axioms are "true" in any sense; this allows =
for=20
parallel mathematical theories built on alternate sets of axioms (see Axi=
omatic=20
set theory and Non-Euclidea=
n=20
geometry for examples).
The word Proof comes from the Latin probare meaning "to =
test".=20
Related modern words are the English "probe", "proboscis=E2=80=9D, =
"probation", and=20
"probability", and the Spanish "probar" (to smell or taste, or (lesser =
use)=20
touch or test)[4].=20
The early use of "probity" was in the presentation of legal evidence. A =
person=20
of authority, such as a nobleman, was said to have probity, whereby the =
evidence=20
was by his relative authority, which outweighed empirical testimony.[5]
[edit]=20
Nature and =
purpose
There are two different conceptions of mathematical proof.[6]=20
The first is an informal proof, a rigorous natural-language expression =
that is=20
intended to convince the audience of the truth of a theorem. Because of =
their=20
use of natural language, the standards of rigor for informal proofs will =
depend=20
on the audience of the proof. In order to be considered a proof, =
however, the=20
argument must be rigorous enough; a vague or incomplete argument is not =
a proof.=20
Informal proofs are the type of proof typically encountered in published =
mathematics. They are sometimes called "formal proofs" because of their =
rigor,=20
but logicians use the term "formal proof" to refer to a different type =
of proof=20
entirely.
In logic, a formal proof is =
not written=20
in a natural language, but instead uses a formal =
language=20
consisting of certain strings of symbols from a fixed alphabet. This =
allows the=20
definition of a formal proof to be precisely specified without any =
ambiguity.=20
The field of proof theory =
studies formal=20
proofs and their properties. Although each informal proof can, in =
theory, be=20
converted into a formal proof, this is rarely done in practice. The =
study of=20
formal proofs is used to determine properties of provability in general, =
and to=20
show that certain u=
ndecidable=20
statements are not provable.
A classic question in philosophy asks whether mathematical proofs are =
analytic =
or synthetic=
. Kant,=20
who introduced the anal=
ytic-synthetic=20
distinction, believed mathematical proofs are synthetic.
Proofs may be viewed as aesthetic objects, admired for their mathematical =
beauty.=20
The mathematician Paul =
Erd=C5=91s was known for=20
describing proofs he found particularly elegant as coming from "The =
Book", a=20
hypothetical tome containing the most beautiful method(s) of proving =
each=20
theorem. The book Proofs from =
THE=20
BOOK, published in 2003, is devoted to presenting 32 proofs its =
editors=20
find particularly pleasing.
[edit]=20
Methods of =
proof
[edit]=20
Direct proof
Main article:
Direct=20
proofIn direct proof[7],=20
the conclusion is established by logically combining the axioms, =
definitions,=20
and earlier theorems. For example, direct proof can be used to establish =
that=20
the sum of two even integers is always=20
even:
- For any two even integers x =
and y we can write x =3D=20
2a and y =3D =
2b for some=20
integers a and b, since both x=20
and y are multiples of 2. But the =
sum x + y =3D 2a + 2b =3D =
2(a +=20
b) is also a multiple of 2, so it is therefore even by=20
definition.
This proof uses definition of even integers, as well as distribution =
law.
[edit]=20
Proof by =
mathematical=20
induction
Main article:
Mathematical=
=20
inductionIn proof by mathematical induction[8],=20
first a "base case" is proved, and then an "induction rule" is used to =
prove a=20
(often infinite) series of =
other=20
cases. Since the base case is true, the infinity of other cases must =
also be=20
true, even if all of them cannot be proved directly because of their =
infinite=20
number. A subset of induction is infinite =
descent.=20
Infinite descent can be used to prove the irrationality=20
of the square root of two.
The principle of mathematical induction states that: Let N =3D =
{ 1, 2,=20
3, 4, ... } be the set of natural numbers and P(n) =
be a=20
mathematical statement involving the natural number n belonging =
to=20
N such that
- (i) P(1) is true, i.e., P(n) is =
true=20
for n =3D 1=20
- (ii) P(n + 1) is true =
whenever=20
P(n) is true, i.e., P(n) is true implies =
that=20
P(n + 1) is true.
Then P(n) is true for all natural numbers =
n.
Mathematicians often use the term "proof by induction" as shorthand =
for a=20
proof by mathematical induction.[9]=20
However, the term "proof by induction" may also be used in logic to mean =
an=20
argument that uses inductive=20
reasoning.
[edit]=20
Proof by=20
transposition
Main article:
Transposition=
=20
(logic)Proof by transposition establishes the conclusion "if p =
then=20
q" by proving the equivalent contrapositive =
statement=20
"if not q then not p".
[edit]=20
Proof by=20
contradiction
Main article:
Proof by=20
contradictionIn proof by contradiction (also known as reductio ad =
absurdum,=20
Latin for "by reduction toward the absurd"), it is shown that if some =
statement=20
were so, a logical contradiction occurs, hence the statement must be not =
so.=20
This method is perhaps the most prevalent of mathematical proofs. A =
famous=20
example of a proof by contradiction shows that ![3D\sqrt{2}=20](3D"http://upload.wikimedia.org/math/e/f/5/ef5590434a387b3c4427e09d5b0=)
=20
is an Irrational =
number:
- Suppose that =20
is a rational number, so
![3D"\sqrt{2}](3D"http://upload.wikimedia.org/math/8/0/3/8039c5b69d82f6fad603d651f00=)
=20
where a and b are non-zero integers with no common factor =
(definition=20
of a rational number). Thus, ![3D"b\sqrt{2}](3D"http://upload.wikimedia.org/math/6/c/0/6c017fd6026a2be3a8288cb8932=)
.=20
Squaring both sides yields 2b2 =3D =
a2. Since=20
2 divides the left hand side, 2 must also divide the right hand side =
(as they=20
are equal and both integers). So a2 is even, which =
implies=20
that a must also be even. So we can write a =3D =
2c, where=20
c is also an integer. Substitution into the original equation =
yields=20
2b2 =3D (2c)2 =3D =
4c2.=20
Dividing both sides by 2 yields b2 =3D =
2c2.=20
But then, by the same argument as before, 2 divides =
b2, so=20
b must be even. However, if a and b are both =
even, they=20
share a factor, namely 2. This contradicts our assumption, so we are =
forced to=20
conclude that =20
is an irrational number.
[edit]=20
Proof by=20
construction
Proof by construction, or proof by example, is the =
construction of a=20
concrete example with a property to show that something having that =
property=20
exists. Joseph =
Liouville, for=20
instance, proved the existence of transcendenta=
l=20
numbers by constructing an explicit =
example.
[edit]=20
Proof by =
exhaustion
Main article:
Proof by=20
exhaustionIn proof by exhaustion, the conclusion is established by =
dividing it=20
into a finite number of cases and proving each one separately. The =
number of=20
cases sometimes can become very large. For example, the first proof of =
the four colour =
theorem=20
was a proof by exhaustion with 1,936 cases. This proof was controversial =
because=20
the majority of the cases were checked by a computer program, not by =
hand. The=20
shortest known proof of the four colour theorem today still has over 600 =
cases.
[edit]=20
Probabilistic =
proof
Main article:
Probabilistic =
methodA probabilistic proof is one in which an example is shown to =
exist,=20
with certainty, by using methods of probability =
theory.=20
This is not to be confused with an argument that a theorem is 'probably' =
true.=20
The latter type of reasoning can be called a 'plausibility argument' and =
is not=20
a proof; in the case of the Collatz =
conjecture it=20
is clear how far that is from a genuine proof.[10]=20
Probabilistic proof, like proof by construction, is one of many ways to =
show existence=20
theorems.
[edit]=20
Combinatorial =
proof
Main article:
Combinatorial=20
proofA combinatorial proof establishes the equivalence of different =
expressions by showing that they count the same object in different =
ways. Often=20
a bijection =
between two=20
sets is used to show that the expressions for their two sizes are equal. =
Alternatively, a d=
ouble=20
counting argument provides two different expressions for the size of =
a=20
single set, again showing that the two expressions are equal.
[edit]=20
Nonconstructive=20
proof
A nonconstructive proof establishes that a certain mathematical =
object=20
must exist (e.g. "Some X satisfies f(X)"), without explaining how such =
an object=20
can be found. Often, this takes the form of a proof by contradiction in =
which=20
the nonexistence of the object is proven to be impossible. In contrast, =
a=20
constructive proof establishes that a particular object exists by =
providing a=20
method of finding it. A famous example of a nonconstructive proof shows =
that=20
there exist two irrational =
numbers=20
a and b such=20
that ab is a rational =
number:
- Either
![3D\sqrt{2}^{\sqrt{2}}=20](3D"http://upload.wikimedia.org/math/c/e/e/cee38f99f384cc23f82b7a2f0d8=)
=20
is a rational number and we are done (take ![3Da=3Db=3D\sqrt{2}=20](3D"http://upload.wikimedia.org/math/8/7/3/873e4f705c98008de708cd7d139=)
),=20
or =20
is irrational so we can write ![3Da=3D\sqrt{2}^{\sqrt{2}}=20](3D"http://upload.wikimedia.org/math/3/b/d/3bde0c4e13c2cee57b9ec04f5ab=)
=20
and ![3Db=3D\sqrt{2}=20](3D"http://upload.wikimedia.org/math/c/7/b/c7b454aae778e1f1ca31425099a=)
.=20
This then gives ![3D"\left](3D"http://upload.wikimedia.org/math/e/6/2/e62273b5738183ebe61f7d57599=)
,=20
which is thus a rational of the form ab.
[edit]=20
Visual proof
=20
Visual proof for the (3, 4, 5) triangle as in the =
Chou Pei Suan =
Ching=20
500=E2=80=93200 BC.
Although not a formal proof, a visual demonstration of a mathematical =
theorem=20
is sometimes called a "proof without=20
words". The picture at right is an example of a historic visual =
proof of the=20
Pythagorean =
theorem=20
in the case of the (3,4,5) triangle.
[edit]=20
Elementary =
proof
Main article:
Elementary =
proofAn elementary proof is a proof which only uses basic techniques. More =
specifically, the term is used in number theory to =
refer to=20
proofs that make no use of complex =
analysis. For=20
some time it was thought that certain theorems, like the prime number=20
theorem, could only be proved using "higher" mathematics. However, =
over=20
time, many of these results have been reproved using only elementary=20
techniques.
[edit]=20
Two-column =
proof
=20
A two-column proof published in =
1913
A particular form of proof using two parallel columns is often used =
in=20
elementary geometry classes.[11]=20
The proof is written as a series of lines in two columns. In each line, =
the left=20
hand column contains propositions (or sometimes called statements), =
while the=20
right hand column contains a brief explanation of how this proposition =
is either=20
an axiom, a hypothesis, or can be obtained from previous lines (or =
sometimes=20
just called "reasons").
[edit]=20
Statistical=20
proofs in pure mathematics
Main article:
Statistical=20
proofThe expression "statistical proof" may be used technically or =
colloquially in=20
areas of pure =
mathematics, such=20
as involving cryptography, chaotic=20
series, and probabilistic or analytic number =
theory.[12][13][14]=20
It is less commonly used to refer to a mathematical proof in the branch =
of=20
mathematics known as mathematica=
l=20
statistics. See also "Statistical proof using data" section below.[4].
[edit]=20
Computer-assisted=20
proofs
Main article:
Computer-as=
sisted=20
proofUntil the twentieth century it was assumed that any proof could, in=20
principle, be checked by a competent mathematician to confirm its =
validity.[3]=20
However, computers are now used both to prove theorems and to carry out=20
calculations that are too long for any human or team of humans to check; =
the=20
first proof of the four color =
theorem is=20
an example of a computer-assisted proof. Some mathematicians are =
concerned that=20
the possibility of an error in a computer program or a run-time error in =
its=20
calculations calls the validity of such computer-assisted proofs into =
question.=20
In practice, the chances of an error invalidating a computer-assisted =
proof can=20
be reduced by incorporating redundancy and self-checks into =
calculations, and by=20
developing multiple independent approaches and programs.
[edit]=20
Undecidable=20
statements
A statement that is neither provable nor disprovable from a set of =
axioms is=20
called undecidable (from those axioms). One example is the parallel =
postulate,=20
which is neither provable nor refutable from the remaining axioms of =
Euclidean=20
geometry.
Mathematicians have shown there are many statements that are neither =
provable=20
nor disprovable in Zermelo=
-Fraenkel=20
set theory with the axiom of choice (ZFC), the standard system of =
set theory=20
in mathematics (assuming that ZFC is consistent); see list=20
of statements undecidable in ZFC.
G=C3=B6del's=20
(first) incompleteness theorem shows that many axiom systems of =
mathematical=20
interest will have undecidable statements.
[edit]=20
Heuristic =
mathematics and=20
experimental mathematics
Main article:
Experiment=
al=20
mathematicsWhile early mathematicians such as Eudoxus of =
Cnidus did=20
not use proofs, from Euclid to the foundation=
al=20
mathematics developments of the late 19th and 20th centuries, proofs =
were an=20
essential part of mathematics.[15]=20
With the increase in computing power in the 1960=E2=80=99s, significant =
work began to be=20
done investigating mathematical=20
objects outside of the proof-theorem framework,[16]=20
in experiment=
al=20
mathematics. Early pioneers of these methods intended the work =
ultimately to=20
be embedded in a classical proof-theorem framework, e.g. the early =
development=20
of fractal =
geometry[17],=20
which was ultimately so embedded.
[edit]=20
Related =
concepts
[edit]=20
Colloquial=20
use of "mathematical proof"
The expression "mathematical proof" is used by lay people to refer to =
using=20
mathematical methods or arguing with mathematical=20
objects, such as numbers, to demonstrate something about everyday =
life, or=20
when data used in an argument are numbers. It is sometime also used to =
mean a=20
"statistical proof" (below), especially when used to argue from =
data.
[edit]=20
Statistical =
proof using=20
data
Main article:
Statistical=20
proof"Statistical proof" from data refers to the application of statistics,=20
data=20
analysis, or Bayesian =
analysis to=20
infer propositions regarding the probability of data. While using=20
mathematical proof to establish theorems in statistics, it is usually =
not a=20
mathematical proof in that the assumpions from which probability=20
statements are derived require empirical evidence from outside =
mathematics to=20
verify. In physics, in addition =
to=20
statistical methods, "statistical proof" can refer to the specialized =
mat=
hematical=20
methods of physics applied to analyze data in a particle =
physics experiment=20
or observational =
study=20
in cosmology. =
"Statistical proof"=20
may also refer to raw data or a convincing diagram involving data, such =
as scatter plots, =
when the=20
data or diagram is adequately convincing without further anaylisis.
[edit]=20
Inductive logic proofs =
and=20
Bayesian analysis
Proofs using inductive =
logic, while=20
considered mathematical in nature, seek to establish propositions with a =
degree of =
certainty,=20
which acts in a similar manner to probability, and =
may be less=20
than one certainty. Bayesian =
analysis=20
establishes assertions as to the degree of a person's subjective =
belief.=20
Inductive logic should not be confused with mathematical=
=20
induction.
[edit]=20
Proofs as mental =
objects
Psychologism views mathematical proofs as psychological or mental =
objects.=20
Mathematician philosophers, such =
as Leibnitz, Frege, and Carnap, =
have=20
attempted to develop a semantics for what they considered to be the language of =
thought,=20
whereby standards of mathematical proof might be applied to empirical =
science.
[edit]=20
Influenc=
e of=20
mathematical proof methods outside mathematics
Philosopher-mathematicians such as Schopenhauer have =
attempted=20
to formulate philosophical arguments in an axiomatic manner, whereby=20
mathematical proof standards could be applied to argumentation in =
general=20
philosophy. Other mathematician-philosophers have tried to use standards =
of=20
mathematical proof and reason, without empiricism, to arrive at =
statements=20
outside of mathematics, but having the certainty of =
propositions=20
deduced in a mathematical proof, such as Descarte=E2=80=99s cogito =
argument.=20
Kant and Frege =
considered=20
mathematical proof to be analytic =
apriori.
[edit]=20
Ending a proof
Sometimes, the abbreviation "Q.E.D." is written to indicate =
the end of=20
a proof. This abbreviation stands for "Quod Erat Demonstrandum", =
which is=20
Latin =
for "that=20
which was to be demonstrated". A more common alternative is to use a =
square=20
or a rectangle, such as =E2=96=A1 or =E2=88=8E, known as a "tombstone" or "halmos". Often, "which =
was to be=20
shown" is verbally stated when writing "QED", "=E2=96=A1", or=20
"=E2=88=8E" in an oral presentation on a =
board.
[edit]=20
See also
[edit]=20
References
- ^ a=20
b=20
Cupillari, Antonella. The Nuts and Bolts of Proofs. Academic =
Press,=20
2001. Page 3.=20
- ^=
=20
Gossett, Eric. Discrete Mathematics with Proof. John Wiley and =
Sons,=20
2009. Definition 3.1 page 86. ISBN=
=20
0470457937=20
- ^ a=20
b=20
The History and Concept of Mathematical Proof, =
Steven G.=20
Krantz. 1. February 5, 2007=20
- ^=
=20
New Shorter Oxford English Dictionary, 1993, OUP, Oxford.=20
- ^=
=20
The Emergence of Probability, Ian Hacking=20
- ^=
=20
Buss, 1997, p. 3=20
- ^=
=20
Cupillari, page 20.=20
- ^=
=20
Cupillari, page 46.=20
- ^=
Proof by induction, University of Warwick Glossary =
of=20
Mathematical Terminology=20
- ^=
=20
While most mathematicians do not think that probabilistic evidence =
ever counts=20
as a genuine mathematical proof, a few mathematicians and philosophers =
have=20
argued that at least some types of probabilistic evidence (such as =
Rabin=E2=80=99s probabilist=
ic=20
algorithm for testing primality) are as good as genuine =
mathematical=20
proofs. See, for example, Davis, Philip J. (1972), "Fidelity in =
Mathematical=20
Discourse: Is One and One Really Two?" American Mathematical =
Monthly=20
79:252-63. Fallis, Don (1997), "The Epistemic Status of Probabilistic =
Proof."=20
Journal of Philosophy 94:165-86.=20
- ^=20
Patricio G. Herbst, Establishing a Custom of Proving in American =
School=20
Geometry: Evolution of the Two-Column Proof in the Early Twentieth =
Century,=20
Educational Studies in Mathematics, Vol. 49, No. 3 (2002), pp. =
283-312,=20
- ^=20
=E2=80=9Cin number theory and commutative algebra... in particular the =
statistical=20
proof of the lemma.=E2=80=9D [1]=20
- ^=20
=E2=80=9CWhether constant =CF=80 (i.e., pi) is normal is a confusing =
problem without any=20
strict theoretical demonstration except for some statistical =
proof=E2=80=9D=E2=80=9D=20
(Derogatory use.)[2]=20
- ^=20
=E2=80=9Cthese observations suggest a statistical proof of Goldbach's =
conjecture with=20
very quickly vanishing probability of failure for large E=E2=80=9D [3]=20
- ^=20
"What to do with the pictures? Two thoughts surfaced: the first was =
that=20
they were unpublishable in the standard way, there were no theorems =
only very=20
suggestive pictures. They furnished convincing evidence for many =
conjectures=20
and lures to further exploration, but theorems were coins of the realm =
ant the=20
conventions of that day dictated that journals only published =
theorems",=20
David Mumford, Caroline Series and David Wright, Indra=E2=80=
=99s=20
Pearls, 2002=20
- ^=20
"Mandelbrot, working at the IBM Research Laboratory, did some =
computer=20
simulations for these sets on the reasonable assumption that, if you =
wanted to=20
prove something, it might be helpful to know the answer ahead of =
time."A Note on the History of Fractals,=20
- ^=20
"=E2=80=A6 brought home again to Benoit [Mandelbrot] that there was =
a =E2=80=98mathematics=20
of the eye=E2=80=99, that visualization of a problem was as valid a =
method as any for=20
finding a solution. Amazingly, he found himself alone with this =
conjecture.=20
The teaching of mathematics in France was dominated by a handful of =
dogmatic=20
mathematicians hiding behind the pseudonym =
=E2=80=98Bourabki=E2=80=99=E2=80=A6 ", Introducing=20
Fractal Geometry, Nigel Lesmoir-Gordon
[edit]=20
Sources
- Polya, G. Mathematics and Plausible Reasoning. Princeton =
University=20
Press, 1954.=20
- Fallis, Don (2002) =E2=80=9CWhat Do Mathematicians=20
Want? Probabilistic Proofs and the Epistemic Goals of =
Mathematicians.=E2=80=9D=20
Logique et Analyse 45:373-88.=20
- Franklin, J. and Daoud, A. Proof in Mathematics: An =
Introduction.=20
Quakers Hill Press, 1996. ISBN=
=20
1-876192-00-3=20
- Solow, D. How to Read and Do Proofs: An Introduction to =
Mathematical=20
Thought Processes. Wiley, 2004. ISBN=
=20
0-471-68058-3=20
- Velleman, D. How to Prove It: A Structured Approach. =
Cambridge=20
University Press, 2006. ISBN=
=20
0-521-67599-5
[edit]=20
External links
v =E2=80=A2 d =E2=80=A2 e Logic |
|
|
History and core=20
topics |
|
| History |
General <=
B>=C2=B7=20
Chinese =
=C2=B7=20
Greek =C2=B7=20
Indian =C2=
=B7=20
Islamic=
|
|
| Core=20
topics |
Reason =C2=B7=20
Philosophical=20
logic =C2=B7 Philosophy=20
of logic =C2=B7 Mathematical=20
logic =C2=B7 Metalogic =C2=
=B7=20
Logic=20
in computer=20
=
science | |
|
|
Key concepts and=20
logics |
|
|
| | <=
/TR>
|
|
|
|
|
|
|
|
|
Languages
- Afrikaans=20
- =D8=A7=D9=84=D8=B9=D8=B1=D8=A8=D9=8A=D8=A9=
=20
- Aragon=
=C3=A9s=20
- B=C3=A2=
n-l=C3=A2m-g=C3=BA=20
- Ca=
tal=C3=A0=20
- =C4=8Ce=
sky=20
- Dansk=20
- Deutsch=20
- E=
spa=C3=B1ol=20
- Esperanto=20
- =D9=81=D8=A7=D8=B1=D8=B3=DB=8C=20
- Fran=C3=A7ais=20
- =ED=95=9C=EA=B5=AD=
=EC=96=B4=20
- Hrvatski=
=20
- Bahasa =
Indonesia=20
- =C3=8Dslenska=20
- Italiano=20
- =D7=A2=
=D7=91=D7=A8=D7=99=D7=AA=20
- =E1=83=A5=E1=83=90=E1=83=A0=E1=83=97=E1=83=A3=E1=83=9A=
=E1=83=98=20
- Latina=
=20
- Lietuvi=C5=
=B3=20
- Lojban=20
- Ma=
gyar=20
- =D0=9C=D0=B0=
=D0=BA=D0=B5=D0=B4=D0=BE=D0=BD=D1=81=D0=BA=D0=B8=20
- Nederlands=20
- =E6=97=A5=E6=9C=AC=
=E8=AA=9E=20
- =E2=80=AANorsk =
(bokm=C3=A5l)=E2=80=AC=20
- =E2=80=AANorsk =
(nynorsk)=E2=80=AC=20
- Plattd=C3=BC=C3=
=BCtsch=20
- Polski =
- Portugu=C3=AA=
s=20
- =D0=A0=D1=83=
=D1=81=D1=81=D0=BA=D0=B8=D0=B9=20
- Simple =
English=20
- Sloven=C5=A1=
=C4=8Dina=20
- =D0=A1=D1=80=
=D0=BF=D1=81=D0=BA=D0=B8=20
/ Srpski=20
- Srpskohrvatski =
/=20
=
=D0=A1=D1=80=D0=BF=D1=81=D0=BA=D0=BE=D1=85=D1=80=D0=B2=D0=B0=D1=82=D1=81=D0=
=BA=D0=B8=20
- Suomi =
- Svenska=20
- Tagalog=20
- =E0=AE=A4=
=E0=AE=AE=E0=AE=BF=E0=AE=B4=E0=AF=8D=20
- T=C3=BCrk=C3=
=A7e=20
- =D0=A3=D0=BA=D1=80=D0=B0=D1=97=D0=BD=D1=81=D1=8C=D0=BA=
=D0=B0=20
- Ti=E1=BA=BFng=20
Vi=E1=BB=87t=20
- =E6=96=
=87=E8=A8=80=20
- =E7=B2=B5=E8=AA=9E=20
- =C5=BDemait=C4=97=C5=A1ka=20
- =E4=B8=AD=E6=96=87=
=20
|
![3D""](3D"http://en.wikipedia.org/wiki/File:Chinese_pythagoras.jpg")
![3D""](3D"http://en.wikipedia.org/wiki/File:Twocolumnproof.png")
![3D"Powered](3D"http://en.wikipedia.org/skins-1.5/common/images/poweredby_mediawik=)
![3D"Wikimedia](3D"http://en.wikipedia.org/images/wikimedia-button.png")