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Discrete Mathematics Lecture 8

Explore the concept of recurrence relations and their applications in discrete mathematics, including famous examples like algebraic and geometric sequences, factorial, Fibonacci numbers, and the Tower of Hanoi problem. Learn how to solve recurrences using iteration, telescoping, range transformation, and domain transformation. Discover second-order homogeneous recurrences and their characteristic equations. Study different classes of functions and understand the relationship between them. Introduce O-notation and its significance in comparing the growth rate of functions.

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Discrete Mathematics Lecture 8

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  1. Discrete MathematicsLecture 8 Alexander Bukharovich New York University

  2. Recursive Sequences • A recurrence relation for a sequence a0, a1, a2, … is a formula that relates each term ak to certain collection of its predecessors. Each recurrence sequence needs initial conditions that make it well-defined • Famous recurrences: algebraic and geometric sequences, factorial, Fibonacci numbers • Tower of Hanoi problem • Compound interest

  3. Exercises • A row in a classroom has n seats. Let sn be the number of ways nonempty sets of students can sit in a row so that no two students occupy adjacent seats. Find recurrence for sn. • In how many ways can one climb n stairs if one is allowed to move to the next stair or jump through one stair? • Show that Fn < 2n • Prove that gcd(Fn+1, Fn) = 1

  4. Solving Recurrences • Iteration method • Telescoping • Range transformation • Domain transformation • Recurrences involving sum

  5. Exercises • Find an explicit formula for: xk = 3xk-1 + k with x1 = 1 wk = wk-2 + k with w1 = 1, w2 = 2 uk = uk-2 * uk-1with u0 = u1 = 2

  6. Second-Order Homogenous Recurrences • Second-order homogeneous relation with constant coefficients is a relation of the form: ak = A * ak-1 + B * ak-2, where A and B are constants • Characteristics equation • Distinct roots case: Fibonacci numbers • Single root case: gambler’s ruin

  7. Classes of Functions • Constants • Polynoms: linear, quadratic • Exponents • Logarithms • Functions in between • Relationship between different classes

  8. O-notation • Function f(n) is of order g(n), written f = O(g), when there exists number M such that there exists number n0 so that for all n > n0 we have f(n) <= M * g(n) • If f is O(g), then g is (f), or in other words, when for all numbers M and for all numbers no, there exists n > n0 such that f(n) > M * g(n) • If f is O(g) and g is O(f), then we say that f is (g) or that f and g are of the same order

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