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This course covered the mathematical topics most directly related to computer science. Topics included: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. The analysis of algorithms requires the ability to count the number of operations in an algorithm. Recursive algorithms in particular depend on the solution to a recurrence equation, and a proof of correctness by mathematical induction. The design of a digital circuit requires the knowledge of Boolean algebra. Software engineering uses sets, graphs, trees and other data structures. Number theory is at the heart of secure messaging systems and cryptography. Logic is used in AI research in theorem proving and in database query systems. Proofs by induction and the more general notions of mathematical proof are ubiquitous in theory of computation, compiler design and formal grammars. Probabilistic notions crop up in architectural trade-offs in hardware design.
Text: Discrete Mathematics and its Applications, Rosen.
Week 1: Introduction, Proofs, Logic, Boolean Algebra and applications, Sets and applications, Basic sums and functions.
Reading: Rosen 1.1-1.8, 3.1-3.2, 5.5, 9.1-9.3 How to Read Mathematics (http://academics.stonehill.edu/compsci/History_Math/math-read.htm), Polya, How to Solve It.
Lecture 1: What kinds of problems are solved in discrete math? What are proofs? Examples of proofs by contradiction, and proofs by induction: Triangle numbers, irrational numbers, and prime numbers. (3.1-3.2)
Lecture 2: Boolean Algebra and formal logic. Applications in algorithms, complexity theory, AI, digital logic design and computer architecture. (1.1-1.2, 9.1-9.3)
Lecture 3: More logic: quantifiers and predicates. Sets, operations on sets, using logic to prove identities on sets. (1.3-1.5)
Lecture 4: Sets. Applications in counting (the inclusion-exclusion theorem), theory of computation and data structures. (5.5)
Lecture 5: Growth rate of functions, Big-O notation, Countability, 1-1 correspondence. Applications to algorithms and theory of computation. (1.6-1.8)
Week 2: Induction, recursion, recurrence equations, graphs.
Reading: Rosen 3.3-3.5, 5.1-5.3, 7.1-7.5
Lecture 1: Basic arithmetic and geometric sums, closed forms. Compound Interest – a simple recurrence. Binary search – recursion, induction and complexity. Towers of Hanoi – recursion, induction, and graphs. (3.3-3.5)
Lecture 2: Chinese rings puzzle – Grey codes, graphs, hypercubes, Hamiltonian and Euler circuits, planar graphs, Euler’s theorem. (7.1-7.5)
Lecture 3: Solving recurrence equations – repeated substitution, the Master Theorem with applications to algorithms, change of variable technique. (5.1, 5.3)
Lecture 4: Solving recurrence equations – guessing and proving correct by induction, linear homogeneous types. The Josephus Problem. (5.2)
Lecture 5: Mathematical induction – a flexible and useful tool. Many examples and the idea of strong induction. (3.2)
Week 3: Counting and discrete probability. Combinations, permutations, pigeonhole principle, inclusion/exclusion revisited.
Reading: Rosen 4.1-4.7, 5.6, How to Read Math (re-read from week 1)
Lecture 1: Combinations and permutations. Pascal’s triangle and binomial coefficients. (4.1, 4.3)
Lecture 2: Counting problems using combinations, distributions and permutations. (4.6)
Lecture 3: The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. (4.2, 4.7, 5.6)
Lecture 4: Discrete probability, the birthday paradox, and many examples. (4.4)
Lecture 5: Conditional probability, and more counting. Generating Functions. (4.5, 5.4)
Week 4: Generating functions, Number theory for cryptography and computer science, equivalence relations, partial orders, trees.
Reading: Rosen 2.1-2.5, 5.4, 6.1-6.6, 8.1-8.2
Lecture 1: Generating functions. (5.4)
Lecture 2: Partial orders, trees and equivalence relations. Applications to algorithms. (8.1-8.2)
Lecture 3: Primes, Greatest Common Divisors and the Euclidean Algorithm. (2.1-2.5)
Lecture 4: The two-jug puzzle as demonstrated by Bruce Willis in Die Hard III.
Lecture 5: Congruences and Fermat’s little theorem. Applications to Cryptography. (6.1-6.6)