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# DISCRETE STRUCTURES MATH 374

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This 2 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 374 at University of South Carolina - Columbia taught by M. Boylan in Fall. Since its upload, it has received 15 views. For similar materials see /class/229517/math-374-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

Math 374 Exam 2 Information Exam 2 will be based on 0 Sections 21 26 31 o The corresponding assigned homework problems see httpwwwmathsceduboylanSCCourses374Sp09374html At minimum you need to understand how to do the homework problems 0 Lecture notes 211 36 Topic List not necessarily comprehensive You will need to know how to de ne vocabulary wordsphrases de ned in class 21 Proof techniques These include 0 Direct proof 0 Contrapositive To prove P a Q it suf ces to give a direct proof of Q a P which is the contrapositive of P a Q Contradiction To prove P a Q it suf ces to give a direct proof of P Q a 0 here 0 denotes a contradiction It is not always clear what the contradiction 0 will be when you begin a proof by contradiction You simply hope to be led to a statement which is clearly false Counterexample To prove that the statement VzPz is false or simply that the state ment Pz is false it suf ces to exhibit a counterexample single zo for which Pz0 is false 22 Induction 0 First principle of induction weak induction To prove that Pn is true Vn 2 no a proof by weak induction proceeds as follows 1 Prove the base case Pn0 2 Perform the induction step Suppose for some k 2 no that Pk is true This is the inductive hypothesis Use it to show that Pk 1 is true 0 Second principle of induction strong induction To prove that Pn is true Vn 2 no a proof by strong induction proceeds as follows 1 Prove the base case or cases There may be more than one base case 2 Perform the induction step Suppose for some k 2 no that Pr is true for all no 3 r g k This is the inductive hypothesis Now use it to prove that Pk 1 is true 23 More on proof of correctness Let 5 denote a loop statement given by while condi tion B is true do P To prove that the loop segment 5 is correct it suf ces by the loop rule to prove the correctness of the triple Q BPQ This triple asserts that the statement Q is a loop invariant a statement which is true before and after every loop iteration Hence to prove that s is correct it suf ces to identify and prove a suitable loop invariant Q Now let Qn be the statement that Q is true after 71 loop iterations To prove that Q is a loop invariant one proves that Qn is true Vn by induction 24 Recursive de nitions Sequences sets and operations may be de ned recursively Recursive de nitions consist of a basis step and a recursive or inductive step For example the Fibonacci sequence is de ned recursively by 0 Fl F2 1 basis 0 Fn 7 l Fn 7 2 Vn 2 3 recursive step 25 Recurrence relations What does it mean for a recurrence relation to be linear to be homogeneous to have order n We solved the following types of recurrence relations 0 Linear lst order recurrence relations 0 Linear homogeneous 2nd order recurrence relations with constant coef cients 5n 01571 7 l 02571 7 2 In this case we study the roots of the characteristic equation pr r2 7 clr 7 02 0 What happens if the roots are real and distinct if there is 1 real repeated root How do you make use of the initial conditions the basis step 26 Analysis of algorithms Count the number of basic operations needed to run an algorithm 31 Sets Subsets proper and improper the empty set power set Venn diagrams union intersection set difference complement cross product set algebra and set identities

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