176
176

Nov 14, 2013
11/13

by
Daniel Kleitman

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In world of mathematics, countless brilliant minds dedicate their lives in an effort to prove the seemingly impossible. Interestingly enough, through the plethora of established proofs which has tremendously impacted the scientific world, a few false proofs have also survived the scrutiny of mathematicians. However, rather than simply dismissing such fallacy proofs as unfortunate mistakes, equally valuable lessons can be learned through the understanding of why such fallacy proofs were able to...

Topics: Maths, Logic, Numbers and Set Theory, Graph Theory, Introduction to Number Systems and Logic, Graph...

Source: http://www.flooved.com/reader/2037

183
183

Nov 14, 2013
11/13

by
John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff

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De�nitions: To describe the complex numbers, we use a formal symbol i representing �_1 ; then a complex number is an expression of the form: a + ib a, b real numbers.

Topics: Maths, Logic, Numbers and Set Theory, Analysis and Calculus, History and Philosophy of Mathematics,...

Source: http://www.flooved.com/reader/1246

136
136

Nov 14, 2013
11/13

by
Kiran S. Kedlaya

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Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Elementary Number Theory,...

Source: http://www.flooved.com/reader/2070

202
202

Nov 14, 2013
11/13

by
Abhinav Kumar

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Continued Fractions - different way to represent real numbers

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Introduction to Number Systems and Logic, Elementary...

Source: http://www.flooved.com/reader/1126

197
197

Nov 14, 2013
11/13

by
Albert R. Meyer

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Why should you care? Indeed, isn�t computer science only about �nite sets? Not exactly. For example, we deal with the set of natural numbers N all the time and it is an in�nite set. In fact, that is why we have induction: to reason about predicates over N. In�nite sets are also important in Part IV of the text when we talk about random variables over potentially in�nite sample spaces. So sit back and open your mind for a few moments while we take a very brieflook at in�nity.

Topics: Maths, Logic, Numbers and Set Theory, Sets, Relations and Functions, Real Numbers, Countability and...

Source: http://www.flooved.com/reader/1747

276
276

Nov 14, 2013
11/13

by
Thomas Ward

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Objectives: (1) A real number is a limit of a sequence of �nite decimal expressions. (2) Two real numbers that are not equal di_er by some positive distance and so must have di_erent decimal expansions. (3) The modulus |�| allows arguments about the distance between numbers to be written precisely. (4) The modulus and quanti�ers allow us to precisely de�ne what it means for a sequence of numbers to converge. (5) An increasing bounded sequence converges.

Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Analysis and Calculus, Real...

Source: http://www.flooved.com/reader/3403

147
147

Nov 14, 2013
11/13

by
Richard Stanley

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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Posets and Zorn�s Lemma, Groups, Hasse Diagrams...

Source: http://www.flooved.com/reader/1045

275
275

Nov 14, 2013
11/13

by
Katrin Wehrheim

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Topics: Maths, Logic, Numbers and Set Theory, Analysis and Calculus, Countability and Uncountability,...

Source: http://www.flooved.com/reader/1273

183
183

Nov 14, 2013
11/13

by
Albert R. Meyer

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A cycle in a digraph is de�ned by a path that begins and ends at the same vertex. This includes the cycle of length zero that begins and ends at the vertex. A directed acyclic graph (DAG) is a directed graph with no positive length cycles.

Topics: Maths, Logic, Numbers and Set Theory, Graph Theory, Posets and Zorn�s Lemma, Set Theory,...

Source: http://www.flooved.com/reader/1725

78
78

Nov 14, 2013
11/13

by
Richard Stanley

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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Mathematics

Source: http://www.flooved.com/reader/1047

192
192

Nov 14, 2013
11/13

by
Abhinav Kumar

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(Chinese Remainder Theorem). If the moduli are coprime in pairs (ie., (mi, mj ) = 1 for i = j), then the system has a unique solution mod m1m2 . . . mk.

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Number Theory, Modular...

Source: http://www.flooved.com/reader/1136

253
253

Nov 14, 2013
11/13

by
Albert R. Meyer

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Euclid�s axiom-and-proof approach, now called the axiomatic method, is the foundation for mathematics today. In fact, just a handful of axioms, collectively called Zermelo-Frankel Set Theory with Choice (ZFC), together with a few logical deduction rules, appear to be suf�cient to derive essentially all of mathematics.

Topics: Maths, Logic, Numbers and Set Theory, Countability and Uncountability, Ordinals and Cardinals,...

Source: http://www.flooved.com/reader/1759

246
246

Nov 14, 2013
11/13

by
Abhinav Kumar

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Euler�s Totient Function:The number of elements in a reduced residue system mod m is called Euler�s totient function: _(m)(ie., the number of positive integers � m and coprime to m)

Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Algebra, Elementary Number...

Source: http://www.flooved.com/reader/1135

158
158

Nov 14, 2013
11/13

by
Abhinav Kumar

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Factorization If n is composite, how do we factor in poly(log n) time. The obvious way is to divide by all, which is O(�n)

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Coding and Cryptography,...

Source: http://www.flooved.com/reader/1137

139
139

Nov 14, 2013
11/13

by
Richard Earl

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Recall that we say a subset H of a group G is a subgroup of G if H is a group in its own right under the restriction of G�s group operation. We write this H � G.There is a simple check � the subgroup test � for determining whether a subset is a subgroup

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Groups, Number Theory,...

Source: http://www.flooved.com/reader/1083

245
245

Nov 14, 2013
11/13

by
Abhinav Kumar

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Binomial Coef�cient: If _ _ C and k is a non-negative integer...

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Number Theory, Modular...

Source: http://www.flooved.com/reader/1134

183
183

Nov 14, 2013
11/13

by
Abhinav Kumar

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Direct formula - start with solution a of f(x) _ 0 mod p, and we want a solution mod p^_.

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Number Theory, Modular...

Source: http://www.flooved.com/reader/1138

309
309

Nov 14, 2013
11/13

by
J. Santos

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Topics: Maths, Logic, Numbers and Set Theory, Sets, Relations and Functions, Countability and...

Source: http://www.flooved.com/reader/1072

174
174

Nov 14, 2013
11/13

by
Daniel Kleitman;Peter Shor

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We will describe several methods for factoring. The first of these is kind of fun, but not the best possible, since it becomes hopeless to use if for factoring numbers above say 10^(50) and probably less; it takes a number of steps of the order of N1/4 to factor N, and the steps are not very simple, though not hard in principle. This method is based on iterating a function, mod N.

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Numerical Analysis, Elementary Number Theory, Number...

Source: http://www.flooved.com/reader/1856

78
78

Nov 14, 2013
11/13

by
Richard Stanley

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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Mathematics

Source: http://www.flooved.com/reader/1044

103
103

Nov 14, 2013
11/13

by
Richard Stanley

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In this section we consider a surprising application of certain adjacency ma_trices to some problems in extremal set theory. An important role will also be played by �nite groups. In general, extremal set theory is concerned with �nding (or estimating) the most or least number of sets satisfying given set-theoretic or combinatorial conditions.

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Mathematics

Source: http://www.flooved.com/reader/1046

281
281

Nov 14, 2013
11/13

by
Laurent Demanet

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Throughout these notes we�ll keep running into Taylor series and Fourier se_ries. It�s important to understand what is meant by convergence of series be_fore getting to numerical analysis proper. These notes are sef-contained, but two good extra references for this chapter are Tao, Analysis I; and Dahlquist and Bjorck, Numerical methods.

Topics: Maths, Logic, Numbers and Set Theory, Analysis and Calculus, Numerical Analysis, Real Numbers,...

Source: http://www.flooved.com/reader/1592

354
354

Nov 14, 2013
11/13

by
Sanjoy Mahajan

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Do you ever walk through a proof, understand each step, yet not believe the theorem, not say �Yes, of course it�s true�? The analytic, logical, sequential approach often does not convince one as well as does a carefully crafted picture. This di_erence is no coincidence. The analytic, sequential portions of our brain evolved with our capacity for language, which is perhaps 105 years old. Our pictorial, Gestalt hardware results from millions of years of evolution of the visual system and...

Topics: Maths, Logic, Numbers and Set Theory, Linear Algebra and Geometry, Analysis and Calculus,...

Source: http://www.flooved.com/reader/1919

152
152

Nov 14, 2013
11/13

by
Albert R. Meyer

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The familiar order relations on numbers have an important additional property: given two different numbers, one will be bigger than the other. Partial orders with this property are said to be total orders.

Topics: Maths, Logic, Numbers and Set Theory, Posets and Zorn�s Lemma, Partially Ordered Sets, Hasse...

Source: http://www.flooved.com/reader/1724

300
300

Nov 14, 2013
11/13

by
Henri Johnston

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These notes are intended to complement rather than replace lectures (for instance, the diligent student should read at least one or two lectures ahead). Indeed, there may well be di_erences between these notes and what I write in lectures. Finally, I would encourage all students to consult the recommended texts, as they contain many more details and examples, and had much more time and e_ort put in to them, than these notes.

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Number Theory, Euclid�s...

Source: http://www.flooved.com/reader/1848

123
123

Nov 14, 2013
11/13

by
Richard Stanley

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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Mathematics

Source: http://www.flooved.com/reader/1043

218
218

Nov 14, 2013
11/13

by
Abhinav Kumar

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Number of primitive roots- suppose that m is an integer such that there is a primitive root g mod m

Topics: Maths, Logic, Numbers and Set Theory, Algebra, Elementary Number Theory, Number Theory,...

Source: http://www.flooved.com/reader/1139

1,578
1.6K

Nov 14, 2013
11/13

by
Peter Ouwehand

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These notes are for a short course in set theory at the undergraduate level at Stellenbosch University. No pretense at orignality is claimed. Though ampli�ed by material from a number of additional sources, the debt to the �rst few chapters of the book Set Theory, by Thomas Jech, Springer 2003, should be easily discernible.

Topics: Maths, Logic, Numbers and Set Theory, Set Theory, Mathematics

Source: http://www.flooved.com/reader/3368

236
236

Nov 14, 2013
11/13

by
Joachim Lambek

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Topics: Maths, Logic, Numbers and Set Theory, Algebra, Real Numbers, Propositional Logic, Set Theory,...

Source: http://www.flooved.com/reader/3381

935
935

Nov 14, 2013
11/13

by
Prof. Stefan Bilaniuk

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A Problem Course in Mathematical Logic is intended to serve as the text for an introduction to mathematical logic for undergraduates with some mathematical sophistication. It supplies definitions, statements of results, and problems, along with some explanations, examples, and hints. The idea is for the students, individually or in groups, to learn the material by solving the problems and proving the results for themselves. The book should do as the text for a course taught using the modified...

Topics: Logic, Numbers and Set Theory, Introduction to Number Systems and Logic, Propositional Logic, Set...

Source: http://www.flooved.com/reader/3492