1 The Foundations: Logic And Proofs 2 Basic Structures: Sets, Functions, Sequences, Sums, And Matrices 3 Algorithms 4 Number Theory And Cryptography 5 Induction And Recursion 6 Counting 7 Discrete Probability 8 Advanced Counting Techniques 9 Relations 10 Graphs 11 Trees 12 Boolean Algebra 13 Modeling Computation A Appendices expand_more
5.1 Mathematical Induction 5.2 Strong Induction And Well-ordering 5.3 Recursive Definitions And Structural Induction 5.4 Recursive Algorithms 5.5 Program Correctness Chapter Questions expand_more
Problem 1E: Findf(1),f(2),f(3), andf(4) iff(n) is defined recursively byf(o) = 1 and forn= o, 1, 2, a)f(n +... Problem 2E: Findf(1),f(2),f(3),f(4), andf(5)iff(n)is defined recursively byf(o)=3and forn=0,1,2,...... Problem 3E: LetP(n) bethestatementthata postage ofncents can be formed using just 3-cent stamps and5-cent... Problem 4E Problem 5E: Determine which amounts of postage can be formed using just 4-cent and 11-cent stamps. Prove your... Problem 6E: Determine which amounts of postage can be formed usingjust 3-cent and 10-cent stamps. Prove your... Problem 7E: e a recursive definition of the sequencean,n=1,2,3,...ifa)an=6n. b)an=2n+1.c)an=10nd)an=5 Problem 8E: Give a recursive definition of the sequence {a),n=1, 2,3, ... if... Problem 9E: Fbe the function such thatF(n) is the sum of the first n positive integers. Give a recursive... Problem 10E: en a recursive definition ofsm(n), the sum of the integermand the nonnegative integern. Problem 11E: e a recursive definition ofPm(n), the product of the integer m and the nonnegative integer n. Problem 12E: Exercises 12—19fnis the nth Fibonacci 12.Prove thatf12+f22+...+fn2=fnfn+1. whennisapositive integer. Problem 13E: Exercises1219fnis the nth Fibonacci number. 13.Prove thatf1+f1+f3+...+f2n1=f2nwhennis a positive... Problem 14E: Exercises 12—l9fnis the nth Fibonacci *14.Show thatfn+1fn1fn2=(1)nwhennis a positive integer. Problem 15E Problem 16E Problem 17E: Exercises 12-19fnis thenthFibonacci number. 17.Determine the number of divisions used by the... Problem 18E: Exercises 12-19fnis thenthFibonacci number. 18. Let A=[1101] Show that An=[fnfn+1fn1fn] whennis a... Problem 19E Problem 20E: e a recursive definition of the if functions max and min so that max(a1,a2,...,an)and... Problem 21E Problem 22E Problem 23E Problem 24E: e a recursive definition of a)the set of odd positive integers. b)the set of positive integer powers... Problem 25E: e a recursive definition of a)the set of even integers. b)the set of positive integers congruent... Problem 26E: Sbe the set of positive integers defined by Basis step:1S. Recursive step:IfnS,then3n+2Sandn2S. a)... Problem 27E: Sbe the set of positive integers defined by Basis step: Recursive step:IfnS,then 3nSandn2S. a)Show... Problem 28E: Sbe the subset of the set of ordered pairs of integers defined recursively byBasis step: (O,O)£S.... Problem 29E: Sbe the subset of the set of ordered pairs of integers defined recursively by JfcsB5fep.,(a,o)GS.... Problem 30E: e a recursive definition of each ofthesesets of ordered pairs of positive integers,[EkPlot tiie... Problem 31E: e arecursive definition of each of these sets of ordered pairs of positiveintegers.Use structural... Problem 32E: ve that in a bit string, the string 01 occurs at most one more time than the string 10. Problem 33E: ine well-formed formulae of sets, variables representing sets, and operators from,,,. Problem 34E Problem 35E: Give a recursive definition of the functionm(s)which equals the smallest digit in a nonempty string... Problem 36E: d the reversal of the following bit strings. a)0101 b)1 1011 c)1000 1001 0111 Problem 37E: e a recursive definition of the reversal of a string. [Hint: First define the reversal of the empty... Problem 38E: structural induction to prove that(w1w2)R=w2Rw1R. Problem 39E Problem 40E: the well-ordermg property to show that ifxandyare real numbers withx< y, then there is arational... Problem 41E: n does a swing belong to eset Aof bit stings defined recursively by A0xlAifxA. whereis the empty... Problem 42E: ursively define the set of bit strings that have more zeros than ones. Problem 43E Problem 44E Problem 45E: structural induction to show thatn(T)>&[I)+inhere lis a full binary tree, n(I) equals tiienumber of... Problem 46E Problem 47E Problem 48E: generalized induction as was doneinExample 13to show that ifamnis defined recursively byan= 5 and... Problem 49E: A partition of a positive integer nis amy to mitenas a sum of positive integers where the order of... Problem 50E Problem 51E: sider the Mowing inductive definition of a version ofAckermann's function.This function was named... Problem 52E Problem 53E Problem 54E: sider the Mowing inductive definition of a version ofAekermann's function.This function was named... Problem 55E: sider the Mowing inductive definition of a version ofAekermann's function.This function was named... Problem 56E Problem 57E: sider the Mowing inductive definition of a version ofAekermann's function.This function was named... Problem 58E Problem 59E Problem 60E Problem 61E Problem 62E: rcises 62-64 deal with iterations of the logarithm function. Let logndenote the logarithm ofnto the... Problem 63E: rcises 62-64 deal with iterations of the logarithm function. Let log n denote the logarithm of n to... Problem 64E Problem 65E Problem 66E: f(n)=n/2.Find a formula forf(k)(n).What is the value of “whennis a positive integer? Problem 67E format_list_bulleted