Week 1: Logic Lecture 1, 8/21 (Sections 1.1 and 1.3)
... But if we assume P1 , P2 are true for the sake of our original argument, then the only way for the modified argument to be valid is if ¬Q is false, namely Q is true. To set up a proof by contradiction, take the negation of the conclusion, add it to the premises, and try to derive something false (a ...
... But if we assume P1 , P2 are true for the sake of our original argument, then the only way for the modified argument to be valid is if ¬Q is false, namely Q is true. To set up a proof by contradiction, take the negation of the conclusion, add it to the premises, and try to derive something false (a ...
Divide and Conquer Algorithms
... (a + bi)(c + di) = ac - bd + (bc + ad)i This representation involves 4 multiplications . The mathematician Carl Friedrich Gauss (1777-1855) noticed that it could be done by using only 3 multiplications as bc + ad = (a + b)(c + d) - ac - bd: ...
... (a + bi)(c + di) = ac - bd + (bc + ad)i This representation involves 4 multiplications . The mathematician Carl Friedrich Gauss (1777-1855) noticed that it could be done by using only 3 multiplications as bc + ad = (a + b)(c + d) - ac - bd: ...
MATH 363 Discrete Mathematics SOLUTIONS : Assignment 3 1
... • All her midterms are going to be on the week (Monday-Friday) before Reading week. Show that she is going to have a day with two midterms. There are 5 boxes (day of the week) in which we want to fit 6 pigeons (exams). By the pigeonhole principle, there is at least one day where the student will hav ...
... • All her midterms are going to be on the week (Monday-Friday) before Reading week. Show that she is going to have a day with two midterms. There are 5 boxes (day of the week) in which we want to fit 6 pigeons (exams). By the pigeonhole principle, there is at least one day where the student will hav ...
Solutions to Homework 6 Mathematics 503 Foundations of
... then there are two integers j and k such that m = 2j + 1 and n = 2k. So m2 − n2 = (2j + 1)2 − 4k 2 = 4j 2 + 4j + 1 − 4k 2 . This is an odd number, so is not divisible by 4. Suppose that m is even and n is odd. If so, then there are two integers j and k such that m = 2j and n = 2k + 1. So m2 − n2 = 4 ...
... then there are two integers j and k such that m = 2j + 1 and n = 2k. So m2 − n2 = (2j + 1)2 − 4k 2 = 4j 2 + 4j + 1 − 4k 2 . This is an odd number, so is not divisible by 4. Suppose that m is even and n is odd. If so, then there are two integers j and k such that m = 2j and n = 2k + 1. So m2 − n2 = 4 ...
A coprimality condition on consecutive values of polynomials
... himself proved, using his already mentioned result from [14], that it cannot be if one take at most 16 consecutive terms. On the other hand, Brauer [3] made connection with his earlier paper [4] on an old problem, studied already by Legendre [12], concerning prime gaps. In fact, Erdős [5] himself a ...
... himself proved, using his already mentioned result from [14], that it cannot be if one take at most 16 consecutive terms. On the other hand, Brauer [3] made connection with his earlier paper [4] on an old problem, studied already by Legendre [12], concerning prime gaps. In fact, Erdős [5] himself a ...
The number 26, between 25 and 27
... say (k, k 0 ) is a solution of the diophantine equation y 3 − x2 = 2 with (x, y) ∈ N2 . In fact, the problem is to resolve this equation. Thus, we will prove that Theorem 1 (Particular case of the Catalan problem) The unique solution of the diophantine equation in N2 y 3 − x2 = 2 ...
... say (k, k 0 ) is a solution of the diophantine equation y 3 − x2 = 2 with (x, y) ∈ N2 . In fact, the problem is to resolve this equation. Thus, we will prove that Theorem 1 (Particular case of the Catalan problem) The unique solution of the diophantine equation in N2 y 3 − x2 = 2 ...
HW 2 Solutions
... where 0 ≤ 3 ≤ 9, then = 10 + 3, for some integer . What does this say about the square of ? c. Prove that if we look at perfect cubes, instead of perfect squares, then the first digit can be anything from 0 to 9. SOLUTION TO #5: (15 pts: a-3, b-8, c-4) a. If we write down the first 10 or so per ...
... where 0 ≤ 3 ≤ 9, then = 10 + 3, for some integer . What does this say about the square of ? c. Prove that if we look at perfect cubes, instead of perfect squares, then the first digit can be anything from 0 to 9. SOLUTION TO #5: (15 pts: a-3, b-8, c-4) a. If we write down the first 10 or so per ...
THE PROBABILITY THAT THE NUMBER OF POINTS ON AN
... Cryptographic and computational applications have recently motivated the study of several questions in the theory of elliptic curves over finite fields. For instance, the analysis of the elliptic curve factoring method leads to estimates ([7], [8]) for the probability that the number of points on an ...
... Cryptographic and computational applications have recently motivated the study of several questions in the theory of elliptic curves over finite fields. For instance, the analysis of the elliptic curve factoring method leads to estimates ([7], [8]) for the probability that the number of points on an ...
Pythagoras` Theorem c =a +b - Strive for Excellence Tutoring
... We can use the following formula to create a Pythagorean Triad. Firstly, we need to find the middle number “m” of a Pythagorean Triad, where “s” is an odd number. The third number can then be found using Pythagoras Theorum. m= ...
... We can use the following formula to create a Pythagorean Triad. Firstly, we need to find the middle number “m” of a Pythagorean Triad, where “s” is an odd number. The third number can then be found using Pythagoras Theorum. m= ...
THE E.IRREGULAR PRIMES
... primes is infinite. The proof is similar to the corresponding proof mentioned above. one might therefore expect that it would also be easy to obtain results on the distribution of .E-irregular primes, by modifying suitable methods used in connection with the ordinary irregular primes. It seems, howe ...
... primes is infinite. The proof is similar to the corresponding proof mentioned above. one might therefore expect that it would also be easy to obtain results on the distribution of .E-irregular primes, by modifying suitable methods used in connection with the ordinary irregular primes. It seems, howe ...
The full Müntz Theorem in C[0,1]
... The original Müntz Theorem proved by Müntz [18] in 1914, by Szász [25] in 1916, and anticipated by Bernstein [3] was only for sequences of exponents tending to infinity. Later works, see, for example, [22] and [19], include the above result, as well as a treatment of the case when {λi }∞ i=1 is a ...
... The original Müntz Theorem proved by Müntz [18] in 1914, by Szász [25] in 1916, and anticipated by Bernstein [3] was only for sequences of exponents tending to infinity. Later works, see, for example, [22] and [19], include the above result, as well as a treatment of the case when {λi }∞ i=1 is a ...
3.3 Proofs Involving Quantifiers 1. In exercise 6 of Section 2.2 you
... Since A ∈ F and every element of F is a subset of every element of G, A ∈ G. Therefore, since A ∈ G and x ∈ A, by the definition 2.3.5., x ∈ ∩G, which is defined since G is not an emptyset. But x was an arbitrary element of ∪F, so this shows that ∪F ⊆ ∩G, as required. 14. In this problem all variabl ...
... Since A ∈ F and every element of F is a subset of every element of G, A ∈ G. Therefore, since A ∈ G and x ∈ A, by the definition 2.3.5., x ∈ ∩G, which is defined since G is not an emptyset. But x was an arbitrary element of ∪F, so this shows that ∪F ⊆ ∩G, as required. 14. In this problem all variabl ...
NOTE ON NORMAL DECIMALS
... Copeland and Erdös [3] have proved that if p i, P2 . . . . is any sequence of positive integers such that, for every 0 < 1, the number of p's up to n exceeds nB if n is sufficiently large, then the infinite decimal • p ip 2p 3 . . . is normal . This includes the result that the decimal formed from t ...
... Copeland and Erdös [3] have proved that if p i, P2 . . . . is any sequence of positive integers such that, for every 0 < 1, the number of p's up to n exceeds nB if n is sufficiently large, then the infinite decimal • p ip 2p 3 . . . is normal . This includes the result that the decimal formed from t ...
Perfect powers in Catalan and Narayana numbers
... Conjecture 1. N (a, b) is never a non-trivial perfect k-th power for k ≥ 3. Conjecture 1 seems to be quite hard. For example, for b = 3 (the case b = 2 follows from some known results), it is related to a generalization of the Catalan’s conjecture by Pillai. However we are able to provide some evide ...
... Conjecture 1. N (a, b) is never a non-trivial perfect k-th power for k ≥ 3. Conjecture 1 seems to be quite hard. For example, for b = 3 (the case b = 2 follows from some known results), it is related to a generalization of the Catalan’s conjecture by Pillai. However we are able to provide some evide ...
Mathematical Induction - Singapore Mathematical Society
... As a research student, I became involved in the application of inductive techniques to optimization problems, developing the ideas expounded in Bellman's book [1]. There are now many applications involving a wide range of mathematical models. Some indication of the growth in this field can be found ...
... As a research student, I became involved in the application of inductive techniques to optimization problems, developing the ideas expounded in Bellman's book [1]. There are now many applications involving a wide range of mathematical models. Some indication of the growth in this field can be found ...
Chapter 3 Review HW
... 9. Maurice joined Costco as a membership salesperson in 2005. The number of memberships he sold increased at an average rate of 56 memberships per year between 2005 and 2009. In 2009 he sold a total of 336 memberships. If the number of memberships Mauricio sold continued to increase at the same rate ...
... 9. Maurice joined Costco as a membership salesperson in 2005. The number of memberships he sold increased at an average rate of 56 memberships per year between 2005 and 2009. In 2009 he sold a total of 336 memberships. If the number of memberships Mauricio sold continued to increase at the same rate ...