Appendix A Infinite Sets
... Imagine for a moment that you are a member of an ancient civilization, one that has not yet developed a counting system. Further imagine that you are a shepherd with a collection of sheep and you want to take them out in the countryside to graze then bring them all back home. How would you be sure t ...
... Imagine for a moment that you are a member of an ancient civilization, one that has not yet developed a counting system. Further imagine that you are a shepherd with a collection of sheep and you want to take them out in the countryside to graze then bring them all back home. How would you be sure t ...
CS311H: Discrete Mathematics Mathematical Proof Techniques
... For all integers n, if n 2 is positive, n is also positive. ...
... For all integers n, if n 2 is positive, n is also positive. ...
mathematics - Textbooks Online
... If zero is one of the given two digits, how many 2 digit numbers can be formed ? Form 2-digit number using the following digits. Write the greater and smaller number. ...
... If zero is one of the given two digits, how many 2 digit numbers can be formed ? Form 2-digit number using the following digits. Write the greater and smaller number. ...
Induction and Recursion - Bryn Mawr Computer Science
... property, then the objects defined by the rule also satisfy the property. Because no objects other than those obtained through the BASE and RECURSION conditions are contained in S, it must be the case that every object in S satisfies the property. Example 21 Show that F (n) < 2n where F (n) is the n ...
... property, then the objects defined by the rule also satisfy the property. Because no objects other than those obtained through the BASE and RECURSION conditions are contained in S, it must be the case that every object in S satisfies the property. Example 21 Show that F (n) < 2n where F (n) is the n ...
SPECIAL PRIME NUMBERS AND DISCRETE LOGS IN FINITE
... the pairs c, d for which the integers (7) and (8) are B-smooth and coprime to the indices, and columns are indexed by the good ideals A and B. Then the left-hand side of (14) equals the product of the row (ycd )cd and the matrix (uS, −vR). Thus each relation (13) gives one row (ycd )cd and congruenc ...
... the pairs c, d for which the integers (7) and (8) are B-smooth and coprime to the indices, and columns are indexed by the good ideals A and B. Then the left-hand side of (14) equals the product of the row (ycd )cd and the matrix (uS, −vR). Thus each relation (13) gives one row (ycd )cd and congruenc ...
Total interval numbers of complete r
... The intersection graph of a family F of sets is the graph obtained by representing each set of F as a vertex and joining two vertices with an edge if their corresponding sets intersect. The family of sets is called an intersection representation of its intersection graph. For an intersection represe ...
... The intersection graph of a family F of sets is the graph obtained by representing each set of F as a vertex and joining two vertices with an edge if their corresponding sets intersect. The family of sets is called an intersection representation of its intersection graph. For an intersection represe ...
Final Exam Review WS
... 67. A restaurant offers 7 entrees and 6 desserts. In how many ways can a person order a twocourse meal? 68. License plates in a particular state display 2 letters followed by 2 numbers. How many different license plates can be manufactured? 69. There are 7 performers who are to present their acts at ...
... 67. A restaurant offers 7 entrees and 6 desserts. In how many ways can a person order a twocourse meal? 68. License plates in a particular state display 2 letters followed by 2 numbers. How many different license plates can be manufactured? 69. There are 7 performers who are to present their acts at ...
ONTOLOGY OF MIRROR SYMMETRY IN LOGIC AND SET THEORY
... Keywords: mirror symmetry, foundations of mathematics, incompleteness of Cantor's set theory, solution of continuum problem, logic and philosophy of science, ontology. In 1900, at the Second International Congress of Mathematicians in Paris, Hilbert formulated his famous 23 open mathematical problem ...
... Keywords: mirror symmetry, foundations of mathematics, incompleteness of Cantor's set theory, solution of continuum problem, logic and philosophy of science, ontology. In 1900, at the Second International Congress of Mathematicians in Paris, Hilbert formulated his famous 23 open mathematical problem ...
Old and new deterministic factoring algorithms
... positive real number, as small as we please, with the implied constant in the Oestimate depending on the choice of .) If an extension of the Riemann Hypothesis were to be proved to hold, then class group methods ([Sha], [Sch]) would factor n in O(n = ) steps. Without using Fourier transforms, the ...
... positive real number, as small as we please, with the implied constant in the Oestimate depending on the choice of .) If an extension of the Riemann Hypothesis were to be proved to hold, then class group methods ([Sha], [Sch]) would factor n in O(n = ) steps. Without using Fourier transforms, the ...
Primes and Greatest Common Divisors
... product. So we have that p divides p1 p2 p3 . . . pk , and p divides q, but that means p divides their difference, which is 1. Therefore p ≤ 1. Contradiction. Therefore there are infinitely many primes. ...
... product. So we have that p divides p1 p2 p3 . . . pk , and p divides q, but that means p divides their difference, which is 1. Therefore p ≤ 1. Contradiction. Therefore there are infinitely many primes. ...
pi, fourier transform and ludolph van ceulen
... 2.1. From ancient times till the 16 th century: The earliest values of including the "Biblical" value of 3, were almost certainly found by measurements. In the Egyptian Rhind Papyrus (about 1650 BC) there is good evidence for 4(8/9)2 =3,16. The first theoretical calculation seems to have been ca ...
... 2.1. From ancient times till the 16 th century: The earliest values of including the "Biblical" value of 3, were almost certainly found by measurements. In the Egyptian Rhind Papyrus (about 1650 BC) there is good evidence for 4(8/9)2 =3,16. The first theoretical calculation seems to have been ca ...
PI, FOURIER TRANSFORM AND LUDOLPH VAN CEULEN
... 2.1. From ancient times till the 16th century: The earliest values of π including the "Biblical" value of 3, were almost certainly found by measurements. In the Egyptian Rhind Papyrus (about 1650 BC) there is good evidence for π≈4(8/9)2 =3,16. The first theoretical calculation seems to have been car ...
... 2.1. From ancient times till the 16th century: The earliest values of π including the "Biblical" value of 3, were almost certainly found by measurements. In the Egyptian Rhind Papyrus (about 1650 BC) there is good evidence for π≈4(8/9)2 =3,16. The first theoretical calculation seems to have been car ...
pi, fourier transform and ludolph van ceulen
... 2.1. From ancient times till the 16th century: The earliest values of π including the "Biblical" value of 3, were almost certainly found by measurements. In the Egyptian Rhind Papyrus (about 1650 BC) there is good evidence for π≈4(8/9)2 =3,16. The first theoretical calculation seems to have been car ...
... 2.1. From ancient times till the 16th century: The earliest values of π including the "Biblical" value of 3, were almost certainly found by measurements. In the Egyptian Rhind Papyrus (about 1650 BC) there is good evidence for π≈4(8/9)2 =3,16. The first theoretical calculation seems to have been car ...
Section 2.5 - Concordia University
... Exercise 2 Cont... Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. d) the real numbers between 0 and 2 Answer: S = (0, 2), S is uncountab ...
... Exercise 2 Cont... Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. d) the real numbers between 0 and 2 Answer: S = (0, 2), S is uncountab ...
The Uniform Continuity of Functions on Normed Linear Spaces
... S, T are real normed spaces, f , f1 , f2 are partial functions from S to T , x1 , x2 are points of S, and Y is a subset of S. Let us consider X, S, T and let us consider f . We say that f is uniformly continuous on X if and only if the conditions (Def. 1) are satisfied. (Def. 1)(i) X ⊆ dom f, and (i ...
... S, T are real normed spaces, f , f1 , f2 are partial functions from S to T , x1 , x2 are points of S, and Y is a subset of S. Let us consider X, S, T and let us consider f . We say that f is uniformly continuous on X if and only if the conditions (Def. 1) are satisfied. (Def. 1)(i) X ⊆ dom f, and (i ...
Medieval Mathematics and Mathematicians
... from this venerated book. Almost all that is known of his life comes from a short biography therein, though he was associated with the court of Frederick II, emperor of the Holy Roman Empire. ”I joined my father after his assignment by his homeland Pisa as an officer in the customhouse located at Bu ...
... from this venerated book. Almost all that is known of his life comes from a short biography therein, though he was associated with the court of Frederick II, emperor of the Holy Roman Empire. ”I joined my father after his assignment by his homeland Pisa as an officer in the customhouse located at Bu ...
Module 3.1 - Discrete Information
... 3:19 - because digits were used for counting 3:21 - when you're little you learn to count on 3:23 - your fingers there are some things in 3:26 - this world that are naturally discreet 3:28 - that's for example money and money can 3:30 - be counted so anything that can be 3:31 - counted on your finge ...
... 3:19 - because digits were used for counting 3:21 - when you're little you learn to count on 3:23 - your fingers there are some things in 3:26 - this world that are naturally discreet 3:28 - that's for example money and money can 3:30 - be counted so anything that can be 3:31 - counted on your finge ...
Counting Derangements, Non Bijective Functions and
... counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalizat ...
... counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalizat ...
Lecture 3. Mathematical Induction
... number 2P − 1 also is prime, but recognized shortly that it is not always true. It is true for p = 2, 3, 5, 7 but ...
... number 2P − 1 also is prime, but recognized shortly that it is not always true. It is true for p = 2, 3, 5, 7 but ...
Diffie-Hellman - SNS Courseware
... Based on Discrete Logarithms Widely used in Security Protocols and Commercial Products Williamson of Britain’s CESG claims to have discovered it several years prior to 1976 ...
... Based on Discrete Logarithms Widely used in Security Protocols and Commercial Products Williamson of Britain’s CESG claims to have discovered it several years prior to 1976 ...
MJ Math 1 Adv - Santa Rosa Home
... Add, subtract, multiply, and divide integers, fractions, and terminating decimals, and perform exponential operations with rational bases and whole number exponents including solving problems in everyday contexts. ...
... Add, subtract, multiply, and divide integers, fractions, and terminating decimals, and perform exponential operations with rational bases and whole number exponents including solving problems in everyday contexts. ...
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying ""smoothly"", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in ""continuous mathematics"" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term ""discrete mathematics."" Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.In the university curricula, ""Discrete Mathematics"" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction to efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in freshmen; as such it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics it is sometimes seen a preparatory course, not unlike precalculus in this respect.The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.