14-Mathematical Induction & Recursive Algorithms

14-Mathematical Induction & Recursive Algorithms - EECS...

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Mathematical Induction & Recursive Algorithms David O. Johnson EECS 210 (Fall 2016) 1 EECS 210 Discrete Structures David O. Johnson Fall 2016
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Reminders Exam 2: Thursday, October 20 More later Homework 4 due: Thursday, October 27 before your lecture Mathematical Induction & Recursive Algorithms David O. Johnson EECS 210 (Fall 2016) 2
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Any Questions? Mathematical Induction & Recursive Algorithms 3 David O. Johnson EECS 210 (Fall 2016)
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Exam 2 Review 10% of your grade. Open book, open note, i.e., anything from the lectures or discussion groups plus any notes you have taken. You may use a laptop or phone to view your notes, slides, supplementary material, and e-book only. Remember: When defining sets, use {}, set builder notation, or capital letters to represent important sets (e.g., Z , ). Probably a little harder than Exam 1. Find the truth set for predicates where the domain is the set of integers (Section 2.1): 6 points Calculate the union, intersection, and difference of sets (Section 2.2): 8 points Calculate the composite of two relations (Section 9.1): 8 points Determine whether relations are reflexive, symmetric, antisymmetric, or transitive (Section 9.1): 8 points Draw a directed graph of a relation (Section 9.3): 8 points Find the transitive closures of a relation in the form of a zero-one matrix (Section 9.4): 8 points Mathematical Induction & Recursive Algorithms David O. Johnson EECS 210 (Fall 2016) 4
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Exam 2 Review Determine whether or not relations represented by zero–one matrices are equivalence relations and why, e.g., it is not reflexive (Section 9.5): 6 points Find maximal elements, minimal elements, least element, greatest element, greatest lower bound, and least upper bound of a poset and subsets of the poset represented by a Hasse diagram (Section 9.6): 6 points Find the composite of two functions (Section 2.3): 8 points Find the value of sums, i.e., ∑ (Section 2.4): 8 points Given a sub-optimal greedy algorithm, show how it would not always produce an optimum solution (Section 3.1): 7 points Give a big-O estimate for a function (Section 3.2, Exercises 25 and 27): 6 points Give a big-O estimate for the number additions used in an algorithm (Section 3.3): 7 points Use mathematical induction to prove a statement P( n ) is true whenever n is a positive integer (Section 5.1): 6 points Mathematical Induction & Recursive Algorithms David O. Johnson EECS 210 (Fall 2016) 5
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Any Questions? Mathematical Induction & Recursive Algorithms 6 David O. Johnson EECS 210 (Fall 2016)
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Strong Induction and Well-Ordering (Section 5.2) Strong Induction Example Proof using Strong Induction Well-Ordering Property Example Proof using Well-Ordering Property Mathematical Induction & Recursive Algorithms 7 David O. Johnson EECS 210 (Fall 2016)
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Strong Induction Last time, we discussed mathematical induction (Section 5.1) and showed how to use it to prove a variety of theorems.
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