The Role of Mathematical Logic in Computer Science and
... Fn (n) is unprovable (FnP (n) is false) P P But D 6= Fm for each m P is not computable So D Provability is not Computable (Gödel Undecidability) ...
... Fn (n) is unprovable (FnP (n) is false) P P But D 6= Fm for each m P is not computable So D Provability is not Computable (Gödel Undecidability) ...
Proofs and Proof Methods
... The Game of Chomp • Chomp is a two-player game played with a set of cookies laid out in an m by n rectangular array. The top left cookie is “poisoned”, and the player who takes it loses. A legal move is to take any cookie, along with all the cookies to the right of it and below it. • We know that o ...
... The Game of Chomp • Chomp is a two-player game played with a set of cookies laid out in an m by n rectangular array. The top left cookie is “poisoned”, and the player who takes it loses. A legal move is to take any cookie, along with all the cookies to the right of it and below it. • We know that o ...
Jacobi`s Two-Square Theorem and Related Identities
... Indeed, I gave such a proof in [4]. The proof there consists of two parts. First, it is shown that ...
... Indeed, I gave such a proof in [4]. The proof there consists of two parts. First, it is shown that ...
CE221_week_1_Chapter1_Introduction
... • Selection problem: you have a group of N numbers and would like to determine the kth largest. • I: read them into an array. Sort them in a decreasing order. Return the kth element. • II: read the first k elements into the array. Sort them in decreasing order. Next read the remaining elements one b ...
... • Selection problem: you have a group of N numbers and would like to determine the kth largest. • I: read them into an array. Sort them in a decreasing order. Return the kth element. • II: read the first k elements into the array. Sort them in decreasing order. Next read the remaining elements one b ...
Chapter 17 Proof by Contradiction
... Proof by contradiction strikes many people as mysterious, because the argument starts with an assumption known to be false. The whole proof consists of building up a fantasy world and then knocking it down. Although the method is accepted as valid by the vast majority of theoreticians, these proofs ...
... Proof by contradiction strikes many people as mysterious, because the argument starts with an assumption known to be false. The whole proof consists of building up a fantasy world and then knocking it down. Although the method is accepted as valid by the vast majority of theoreticians, these proofs ...
Chapter 17 Proof by Contradiction
... Proof by contradiction strikes many people as mysterious, because the argument starts with an assumption known to be false. The whole proof consists of building up a fantasy world and then knocking it down. Although the method is accepted as valid by the vast majority of theoreticians, these proofs ...
... Proof by contradiction strikes many people as mysterious, because the argument starts with an assumption known to be false. The whole proof consists of building up a fantasy world and then knocking it down. Although the method is accepted as valid by the vast majority of theoreticians, these proofs ...
HOMEWORK 2 1. P63, Ex. 1 Proof. We prove it by contradiction
... number r such that r2 = 3 and r > 0. Since r is a rational number, then there exists r = pq such that (p, q) = 1, where the notation (a, b) = 1 means that the greatest common divisor of a and b is 1. Then p2 = 3q 2 . That is to say, p2 is a multiple of 3. We may classify p in the following cases, p ...
... number r such that r2 = 3 and r > 0. Since r is a rational number, then there exists r = pq such that (p, q) = 1, where the notation (a, b) = 1 means that the greatest common divisor of a and b is 1. Then p2 = 3q 2 . That is to say, p2 is a multiple of 3. We may classify p in the following cases, p ...
STRONG LAW OF LARGE NUMBERS WITH CONCAVE MOMENTS
... (i) We used only a very special case of the LLN of [3], which applies to groupvalued random variables. Theorem 1 holds indeed also in that setting with identical proof, but can immediately be reduced to the real-valued case. (ii) The point of the present note is that the LLN of [3] brings new insigh ...
... (i) We used only a very special case of the LLN of [3], which applies to groupvalued random variables. Theorem 1 holds indeed also in that setting with identical proof, but can immediately be reduced to the real-valued case. (ii) The point of the present note is that the LLN of [3] brings new insigh ...
Lemma 3.3
... Proof. The proof we present is elementary, but rather long. We decided to present most of the details, including a double induction, because this technique can be used to prove other results. Note that the left hand side of (1) is an odd number, which implies that the numbers of even terms in the nu ...
... Proof. The proof we present is elementary, but rather long. We decided to present most of the details, including a double induction, because this technique can be used to prove other results. Note that the left hand side of (1) is an odd number, which implies that the numbers of even terms in the nu ...
Induction
... Prove by induction that every integer greater than or equal to 2 can be factored into primes. The statement P(n) is that an integer n greater than or equal to 2 can be factored into primes. 1. Base Case : Prove that the statement holds when n = 2 We are proving P(2). 2 itself is a prime number, so t ...
... Prove by induction that every integer greater than or equal to 2 can be factored into primes. The statement P(n) is that an integer n greater than or equal to 2 can be factored into primes. 1. Base Case : Prove that the statement holds when n = 2 We are proving P(2). 2 itself is a prime number, so t ...
Section 1.4 Mathematical Proofs
... legend was made at sea, the Pythagoreans considered this heresy and threw him overboard. So much for making one of the greatest mathematical discoveries of all time. We now prove a famous theorem goes back to 300 B.C to the great Greek mathematican Euclid of Alexandra (present day Egypt) and relies ...
... legend was made at sea, the Pythagoreans considered this heresy and threw him overboard. So much for making one of the greatest mathematical discoveries of all time. We now prove a famous theorem goes back to 300 B.C to the great Greek mathematican Euclid of Alexandra (present day Egypt) and relies ...
On mathematical induction
... proposition Pn stated that xn = 2 22n+1 . In our course usually a proposition will be some sort of simple mathematical statement which is either true or false. If the idea of having infinitely many propositions confuse you, just think to a proposition which involves functions of integer numbers, for ...
... proposition Pn stated that xn = 2 22n+1 . In our course usually a proposition will be some sort of simple mathematical statement which is either true or false. If the idea of having infinitely many propositions confuse you, just think to a proposition which involves functions of integer numbers, for ...
Direct Proof and Counterexample II - H-SC
... Let Q denote the set of rational numbers. An irrational number is a number that is not rational. We will assume that there exist irrational ...
... Let Q denote the set of rational numbers. An irrational number is a number that is not rational. We will assume that there exist irrational ...
hilbert theorem on lemniscate and the spectrum of the perturbed shift
... with c := maxf 1=g. So, the condition z 2 l~a implies z 2 s(T~a), and s(Ta ) = l~a = s(Ta ): End of the proof. Remarks. 1. Every lemniscate (1) is the spectrum of some operator Tra . For example, we can choose a = afeg~a with ~a = fz1 z2 ::: zN g. Then a(n) is a periodic function. ...
... with c := maxf 1=g. So, the condition z 2 l~a implies z 2 s(T~a), and s(Ta ) = l~a = s(Ta ): End of the proof. Remarks. 1. Every lemniscate (1) is the spectrum of some operator Tra . For example, we can choose a = afeg~a with ~a = fz1 z2 ::: zN g. Then a(n) is a periodic function. ...
Lecture Notes - jan.ucc.nau.edu
... 1. Assume that there are a finite number of primes. 2. Then there is a largest prime, p. Consider the number q = (2x3x5x7x...xp)+1. q is one more than the product of all primes up to p. q > p. And, q is not divisible by any prime up to p. For example, it is not divisible by 7, as it is one more than ...
... 1. Assume that there are a finite number of primes. 2. Then there is a largest prime, p. Consider the number q = (2x3x5x7x...xp)+1. q is one more than the product of all primes up to p. q > p. And, q is not divisible by any prime up to p. For example, it is not divisible by 7, as it is one more than ...
Lecture slides (full content)
... # according course info sheet then you are 3% below average in term of final grade # 6%x50%=3% then it is below are the acceptable margin of error # 5% in physics then it is totally acceptable then it is not bad even if you left everything blank and others did well! ...
... # according course info sheet then you are 3% below average in term of final grade # 6%x50%=3% then it is below are the acceptable margin of error # 5% in physics then it is totally acceptable then it is not bad even if you left everything blank and others did well! ...
Document
... • Now, just from this, what do we know about x and y? You should think back to the definition of rational: • … Since x is rational, we know (from the very definition of rational) that there must be some integers i and j such that x = i/j. So, let ix,jx be such integers … • We give them unique names ...
... • Now, just from this, what do we know about x and y? You should think back to the definition of rational: • … Since x is rational, we know (from the very definition of rational) that there must be some integers i and j such that x = i/j. So, let ix,jx be such integers … • We give them unique names ...
![Q1 [20 points] Q2 [20 points]](http://s1.studyres.com/store/data/012801942_1-7e7115778c021991ef6f06b5ae7330d9-300x300.png)
Computability - Homepages | The University of Aberdeen
... n differs from the i-th number on the i-th digit Which of the enumerated numbers is n? • n is not a1 because of a11 • n is not a2 because of a22 Etc. So, n is not in the list! Consequently, the enumeration was not complete after all ...
... n differs from the i-th number on the i-th digit Which of the enumerated numbers is n? • n is not a1 because of a11 • n is not a2 because of a22 Etc. So, n is not in the list! Consequently, the enumeration was not complete after all ...
SESSION 1: PROOF 1. What is a “proof”
... is a big question, which I won’t pretend to answer; instead, I will outline a few reasons why I care about proof. • A proof of a mathematical statement is absolute; there is no exception to the rule. Such absolute statements are wonderful and do not exist in any other field of study! • Mathematics i ...
... is a big question, which I won’t pretend to answer; instead, I will outline a few reasons why I care about proof. • A proof of a mathematical statement is absolute; there is no exception to the rule. Such absolute statements are wonderful and do not exist in any other field of study! • Mathematics i ...
An Invitation to Proofs Without Words
... What are “proofs without words”? As you will see from this article, the question does not have a simple, concise answer. Generally, proofs without words are pictures or diagrams that help the reader see why a particular mathematical statement may be true, and also to see how one might begin to go ab ...
... What are “proofs without words”? As you will see from this article, the question does not have a simple, concise answer. Generally, proofs without words are pictures or diagrams that help the reader see why a particular mathematical statement may be true, and also to see how one might begin to go ab ...
Induction
... But this directly contradicts the fact that P(m − 1) =⇒ P(m). It may seem as though we just proved the induction axiom. But what we have actually done is to show that the induction axiom follows from another axiom, which was used implicitly in defining “the first m for which P(m) is false.” We note ...
... But this directly contradicts the fact that P(m − 1) =⇒ P(m). It may seem as though we just proved the induction axiom. But what we have actually done is to show that the induction axiom follows from another axiom, which was used implicitly in defining “the first m for which P(m) is false.” We note ...
Mathematical induction Elad Aigner-Horev
... Theorem 2.2. (Strong/Complete mathematical induction) Let S(n) denote a mathematical statement that depends on n ∈ Z+ . In addition, let n0 , n1 ∈ Z+ satisfy n0 ≤ n1 . (Base) If S(n0 ), S(n0 + 1), . . . , S(n1 ) are all true; and (Step) if whenever S(n0 ), S(n0 + 1), . . . , S(k − 1), S(k) are true ...
... Theorem 2.2. (Strong/Complete mathematical induction) Let S(n) denote a mathematical statement that depends on n ∈ Z+ . In addition, let n0 , n1 ∈ Z+ satisfy n0 ≤ n1 . (Base) If S(n0 ), S(n0 + 1), . . . , S(n1 ) are all true; and (Step) if whenever S(n0 ), S(n0 + 1), . . . , S(k − 1), S(k) are true ...