Comparing sizes of sets
... The set of all strings (over any alphabet) is countable Recall: An alphabet is a finite set of symbols, and for any alphabet A, A∗ is the set of all strings over A. If A is empty, then A∗ = If A is a singleton, then it is still easy to see that A∗ is countable. Indeed, take f : A∗ → N s.t. for all ...
... The set of all strings (over any alphabet) is countable Recall: An alphabet is a finite set of symbols, and for any alphabet A, A∗ is the set of all strings over A. If A is empty, then A∗ = If A is a singleton, then it is still easy to see that A∗ is countable. Indeed, take f : A∗ → N s.t. for all ...
Unit B391/01 – Sample scheme of work and lesson plan
... 1 – Solve problems using mathematical skills select and use suitable problem solving strategies and efficient techniques to solve numerical problems identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting parti ...
... 1 – Solve problems using mathematical skills select and use suitable problem solving strategies and efficient techniques to solve numerical problems identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting parti ...
Open Problems in Mathematics - School of Mathematical Sciences
... 4 So you’ll understand that the distributive property has nothing to do with real estate. 3 Because solving word problems could lead to solving the world’s problems. 2 To learn that Dewey didn’t invent the decimal. 1 If you can’t count to a million, how will you know if you’ve become a millionaire? ...
... 4 So you’ll understand that the distributive property has nothing to do with real estate. 3 Because solving word problems could lead to solving the world’s problems. 2 To learn that Dewey didn’t invent the decimal. 1 If you can’t count to a million, how will you know if you’ve become a millionaire? ...
Independent domination in graphs: A survey and recent results
... the following bound for trees, which was originally conjectured by McFall and Nowakowski [87]. Theorem 2.14 ([40]). If T is a tree with n vertices and ℓ leaves, then i(G) ≤ (n + ℓ)/3. The bound in Theorem 2.14 is achieved, for example, by the path P3k+1 when k ≥ 1. Other families achieving equality ...
... the following bound for trees, which was originally conjectured by McFall and Nowakowski [87]. Theorem 2.14 ([40]). If T is a tree with n vertices and ℓ leaves, then i(G) ≤ (n + ℓ)/3. The bound in Theorem 2.14 is achieved, for example, by the path P3k+1 when k ≥ 1. Other families achieving equality ...
Unit A502/01 - Sample scheme of work and lesson plan booklet (DOC, 4MB)
... plot graphs of functions in which y is given explicitly in terms of x, or implicitly, where no ...
... plot graphs of functions in which y is given explicitly in terms of x, or implicitly, where no ...
Math - Humboldt Community School District
... Solves real world problems by using a “guess and test” strategy Uses exponents to simplify numbers in scientific notation Solves one and two step equations and inequalities using basic arithmetic operations including combining like terms Correctly simplifies variable expressions ...
... Solves real world problems by using a “guess and test” strategy Uses exponents to simplify numbers in scientific notation Solves one and two step equations and inequalities using basic arithmetic operations including combining like terms Correctly simplifies variable expressions ...
Unit B392/02 – Sample scheme of work and lesson plan
... select and use suitable problem solving strategies and efficient techniques to solve numerical problems identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches break down a complex calcu ...
... select and use suitable problem solving strategies and efficient techniques to solve numerical problems identify what further information may be required in order to pursue a particular line of enquiry and give reasons for following or rejecting particular approaches break down a complex calcu ...
Lecture 2
... There are many other orders to generate all permutations, different from the lexicographic order. Often, we want the fast generation of all permutations: B This means to generate very fast the next permutation from the previous one. B In 1963, Heap discovered an algorithm that generates the next per ...
... There are many other orders to generate all permutations, different from the lexicographic order. Often, we want the fast generation of all permutations: B This means to generate very fast the next permutation from the previous one. B In 1963, Heap discovered an algorithm that generates the next per ...
Those Incredible Greeks! - The Saga of Mathematics: A Brief History
... commercial arithmetic. Coinage in precious metals was invented around 700 BC and gave rise to a money economy based not only on agriculture but also on movable goods. This brought Magna Greece (“greater Greece”) prosperity. ...
... commercial arithmetic. Coinage in precious metals was invented around 700 BC and gave rise to a money economy based not only on agriculture but also on movable goods. This brought Magna Greece (“greater Greece”) prosperity. ...
Ramsey theory - UCSD Mathematics
... 2.4. Graph Ramsey theory Because of the early realization of the difficulty in obtaining sharp results for the classical Ramsey numbers, focus turned to the general study of the numbers r(G, H), for arbitrary graphs (as opposed to complete graphs). When G = H, we write r(G) = r(G, G). There was an i ...
... 2.4. Graph Ramsey theory Because of the early realization of the difficulty in obtaining sharp results for the classical Ramsey numbers, focus turned to the general study of the numbers r(G, H), for arbitrary graphs (as opposed to complete graphs). When G = H, we write r(G) = r(G, G). There was an i ...
Must All Good Things Come to an End?
... Arabic mathematicians, in particular al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. Thābit ibn Qurra undertook both theoretical ...
... Arabic mathematicians, in particular al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. Thābit ibn Qurra undertook both theoretical ...
bsccsit-com_discrete_structure
... represent like “ some students of CDCSIT take graphics class”. The existential quantification is the disjunction of all the propositions that are obtained by assigning the values of the variable from the universe of discourse. So the above example is equivalent to P(ram) ∨ P(shyam) ∨ P(hari) ∨ P(sit ...
... represent like “ some students of CDCSIT take graphics class”. The existential quantification is the disjunction of all the propositions that are obtained by assigning the values of the variable from the universe of discourse. So the above example is equivalent to P(ram) ∨ P(shyam) ∨ P(hari) ∨ P(sit ...
CSE 1400 Applied Discrete Mathematics Permutations
... For small sets each permutation can be listed. Let A be a set with cardinality |A| = n. There are n factorial different permutations of the elements in A. Figure 1 shows the 3! = 6 permutations of the elements in {0, 1, 2} written in cyclic notation. The permutations on {0, 1, 2, 3} can be defined r ...
... For small sets each permutation can be listed. Let A be a set with cardinality |A| = n. There are n factorial different permutations of the elements in A. Figure 1 shows the 3! = 6 permutations of the elements in {0, 1, 2} written in cyclic notation. The permutations on {0, 1, 2, 3} can be defined r ...
GRAPHS WITH EQUAL DOMINATION AND INDEPENDENT
... Since for every vertex v ∈ V (G)−S, there exists a vertex u ∈ S such that v is adjacent to u, it follows that S is a dominating set of G. Moreover, the above set S is an independent set of G because the vertices in S are pairwise non-adjacent vertices. Therefore, the above set S is an independent do ...
... Since for every vertex v ∈ V (G)−S, there exists a vertex u ∈ S such that v is adjacent to u, it follows that S is a dominating set of G. Moreover, the above set S is an independent set of G because the vertices in S are pairwise non-adjacent vertices. Therefore, the above set S is an independent do ...
10-01-2014 Dear Teachers,
... [7] H. Davenport : The higher arithmetic Cambridge Univ.Press, Sixth Edn. (1992) [8] Kenneth H Rosen : Elementary Number Theory and its applications Addison Wesley Pub Co., 3rd Edn., (1993) [9] G.H. Hardy & E M Wright : Introduction to the theory of numbers Oxford International Edn ...
... [7] H. Davenport : The higher arithmetic Cambridge Univ.Press, Sixth Edn. (1992) [8] Kenneth H Rosen : Elementary Number Theory and its applications Addison Wesley Pub Co., 3rd Edn., (1993) [9] G.H. Hardy & E M Wright : Introduction to the theory of numbers Oxford International Edn ...
The Mathematics of Harmony: Clarifying the Origins and
... possess mathematical beauty." This inscription is the famous Principle of Mathematical Beauty that Dirac developed during his scientific life. No other modern physicist has been preoccupied with the concept of beauty more than Dirac. Thus, according to Dirac, the Principle of Mathematical Beauty is ...
... possess mathematical beauty." This inscription is the famous Principle of Mathematical Beauty that Dirac developed during his scientific life. No other modern physicist has been preoccupied with the concept of beauty more than Dirac. Thus, according to Dirac, the Principle of Mathematical Beauty is ...
Order date - Calicut University
... (ii) Functional Analysis is made into a single paper (iii) A new course “Multivariable Calculus and Geometry” is introduced (iv) Differential Geometry paper moved to the elective papers (v) ODE and Calculus of Variation paper is moved to the second semester (vi) Number Theory paper is in the first s ...
... (ii) Functional Analysis is made into a single paper (iii) A new course “Multivariable Calculus and Geometry” is introduced (iv) Differential Geometry paper moved to the elective papers (v) ODE and Calculus of Variation paper is moved to the second semester (vi) Number Theory paper is in the first s ...
Math 7 Scope and Sequence By
... Objectives/concepts TEKS Topics (not in sequential order) Suggested Assessments Resources ...
... Objectives/concepts TEKS Topics (not in sequential order) Suggested Assessments Resources ...
Artificial Intelligence Methods Bayesian networks
... and hence for compact specification of full joint distributions Syntax: a set of nodes, one per variable a directed, acyclic graph (link ≈ “directly influences”) a conditional distribution for each node given its parents: P(Xi|P arents(Xi)) In the simplest case, conditional distribution represented ...
... and hence for compact specification of full joint distributions Syntax: a set of nodes, one per variable a directed, acyclic graph (link ≈ “directly influences”) a conditional distribution for each node given its parents: P(Xi|P arents(Xi)) In the simplest case, conditional distribution represented ...
2017 Discrete Math Pacing Guide
... Huntsville City Schools 2016 – 2017 Pacing Guide Discrete Math Twelfth Grade Math Practices: The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and pro ...
... Huntsville City Schools 2016 – 2017 Pacing Guide Discrete Math Twelfth Grade Math Practices: The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and pro ...
Fibonacci numbers, alternating parity sequences and
... We are particularly concerned with the set Tn whose elements are the n × n tridiagonal doubly stochastic matrices, which is a face of the Birkhoff polytope. The facial structure of n has been the object of a systematic study in the series of papers [3–6], and also in [2,8]. However, the tridiagonal ...
... We are particularly concerned with the set Tn whose elements are the n × n tridiagonal doubly stochastic matrices, which is a face of the Birkhoff polytope. The facial structure of n has been the object of a systematic study in the series of papers [3–6], and also in [2,8]. However, the tridiagonal ...
Proof
... The fallacy of (explicitly or implicitly) assuming the very statement you are trying to prove in the course of its proof. Example Prove that an integer n is even, if n2 is even. Attempted Proof. “Assume n2 is even. Then n2 = 2k for some integer k. Dividing both sides by n gives n = (2k )/n = 2(k/n). ...
... The fallacy of (explicitly or implicitly) assuming the very statement you are trying to prove in the course of its proof. Example Prove that an integer n is even, if n2 is even. Attempted Proof. “Assume n2 is even. Then n2 = 2k for some integer k. Dividing both sides by n gives n = (2k )/n = 2(k/n). ...
No Matter How You Slice It. The Binomial Theorem and - Beck-Shop
... In the last chapter, we started developing enumerative techniques by finding formulae that covered six basic situations. We will continue in that direction in Chapter 5. Now, however, we take a break and discuss the binomial and the multinomial theorems, as well as several important identities on bi ...
... In the last chapter, we started developing enumerative techniques by finding formulae that covered six basic situations. We will continue in that direction in Chapter 5. Now, however, we take a break and discuss the binomial and the multinomial theorems, as well as several important identities on bi ...
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.