1 / 136

461191 Discrete Mathematics Lecture 4: Induction and Recursion

461191 Discrete Mathematics Lecture 4: Induction and Recursion. San Ratanasanya CS, KMUTNB. Today’s Topics. Last week review Administrivia Proof Review Mathematical Induction Recursive Definitions Recursive Algorithms Program Correctness. Last week review. Algorithms,

Download Presentation

461191 Discrete Mathematics Lecture 4: Induction and Recursion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 461191 Discrete MathematicsLecture 4: Induction and Recursion San Ratanasanya CS, KMUTNB

  2. Today’s Topics • Last week review • Administrivia • Proof Review • Mathematical Induction • Recursive Definitions • Recursive Algorithms • Program Correctness

  3. Last week review Algorithms, The Integers, and Matrices

  4. Algorithm is a finite set of precise instructionsfor performing a computation or for solving a problem.

  5. Pseudocode Algorithm 1 Finding the maximum element in a Finite Sequence. procedure max (a1, a2,…, an : integers) max := a1 for i := 2 to n if max < ai then max := ai {max is the largest element}

  6. Algorithm 2The Linear Search Algorithm procedure linear search (x : integer; a1, a2,…, an :distinct integers) i := 1 while (i ≤ n and x ≠ ai) i := i + 1 if i ≤n then location := i else location := 0 {location is the subscript of term that equals x, or is 0 if not found}

  7. Algorithm 3Binary Search Algorithm procedure binary search (x : integer; a1, a2,…, an : increasing integers) i := 1 {i is left endpoint of search interval} j := n {j is right endpoint of search interval} while i < j begin m :=  (i + j)/2  if x > am then i := m + 1 else j := m end if x = ai then location := i else location := 0 {location is the subscript of term equal to x, or 0 if x is not found}

  8. Pseudocodes Procedurebubblesort(a1,…,an: real numbers with n  2) for i := 1 to n-1 for j := 1 to n-1 if aj > aj+1then interchange aj and aj+1 {a1,…,an isin increasing order}

  9. The growth of functions DEFINITION 1 : Let f and g be functions from the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) if there are constants C and k such that, |f(x)| ≤ C |g(x)| when x > k

  10. Example 1 : show that f(x) = x2+2x+1 is O(x2) case 1 0 ≤ x2+2x+1 ≤ x2+2x2+x2 = 4x2, when x >1 So f(x) = O(x2) , when C = 4 and k = 1 Ans case 2 when x >2 , 2x < x2 0 ≤ x2+2x+1 ≤ x2+x2+x2 = 3x2,when x >2 so f(x) = O(x2) when C = 3 and k = 2 Ans In this example, g(x) = O(f(x)) since x2 ≤ x2+2x+1 so, we say that f(x) and g(x) are of the same order

  11. THEOREM 1 : Let f(x) = anxn+an-1xn-1+…+a1x+a0 where a0,a1,…an are real numbers.Then f(x) is O(xn) Proof : |f(x)| = | anxn+an-1xn-1+…+a1x+a0 | ≤ |an|xn+|an-1|xn-1+…+|a1|x+|a0| = xn (|an|+|an-1|x-1+…+ |a1|x(n-1)+ |a0|x-n-1) ≤ xn (|an|+|an-1|+…+ |a1|+ |a0|) This shows that |f(x)| ≤ C xn when C = (|an|+|an-1|+…+ |a1|+ |a0| and x > 1.

  12. Example 2 : 1+2+3+…+n = O(?) ≤ n +n +…+n = n × n = n2 = O(n2) , C = 1 , k = 1 Example 3 : f(n) = n! = O(?) = 1 * 2 * 3 * …* n ≤ n * n * n * …* n = nn ≤ C |g(n)| = O(nn), C = k = 1

  13. Since… …n! ≤ nn log n! ≤ log nn = n log n log n! = O(n log n) ...n < 2n n = O(2n) , k = C = 1 ...log n < n log n = O(n) , k = C = 1 ...logbn = log n/log b < n/log b logbn = O(n) , k = 1 , C = 1/log b

  14. THEOREM 2 : If f1(x) = O(g1(x)) and f2(x) = O(g2(x))then (f1+f2)(x) = O(max(g1(x), g2(x))) since |f1(x)| ≤ C1 |g1(x)| , when x > k1 and |f2(x)| ≤ C2 |g2(x)| , when x > k2 |(f1+f2)(x)| = |f1(x) + f2(x)| ≤ |f1(x)| + |f2(x)| |f1(x)| + |f2(x)| < C1|g1(x)| + C2|g2(x)| ≤ C1|g(x)| + C2|g(x)| = (C1+C2) |g(x)| = C |g(x)| when C = C1+C2 , and g(x) = max|(g1(x),g2(x)| , k = max(k1,k2)

  15. THEOREM 3 : Let f1(x) = O(g1(x)) and f2(x) = O(g2(x)) (f1 f2)(x) = O( g1(x) g2(x) ) Example 4 : f(n) = 3n log (n!)+(n2+3) log n = O(?), where n = positive integer ≤ O(n) O(n log n) + O(2n2) O(log n) ≤ O(n2 log n) + O((2n2) log n) = O(3n2 log n) = O(n2 log n) Example 5 : f(x) = (x+1) log (x2+1) + 3x2 = O(?) ==================================================== Commonly Estimated Functions : 1, log n , n , n log n , n2 , 2n , n!

  16. Complexity Analysis of Algorithms Computational Complexity: 1) Time Complexity: An analysis of the time required to solve a problem. Time complexity is described in terms of the number of operations required instead of actual computer time, such as comparison, addition, subtraction, multiplication, division, etc. There are 3 possible cases in time complexity best case, average case, and worst case

  17. 2) Space Complexity: An analysis of the computer memory required to solve a problem. space complexity are tied in with the particular data structures used to implement the algorithm.

  18. Example 1 Describe the time complexity of Algorithm 1 of Section 2.1 for finding the maximum element in a set. Solution: “The number of comparisons will be used as the measurement of the time complexity of the algorithm, since comparisons are the basic operation used.” • when i = n + 1, there are exactly 2(n - 1) + 1 = 2n - 1 comparisons

  19. Example 2 Describe the time complexity of linear search algorithm.

  20. n = 2k or 2k < n < 2k+1 2k elements in the first loop 2k-1 elements in the second loop : : 21 elements in the k-1th loop 20 elements in the kth loop At most 2k+2 comparisons = (2 log n ) + 2 = O(log n) k loops

  21. Example 3 Describe the time complexity of the binary search algorithm. Solution: For simplicity, assume there are n = 2k elements in the list a1, a2,…,an, where k is a non-negative integer. Note that k = log n (If n, the number of elements in the list, is not a power of 2, the list can be considered as part of a larger list with 2k + 1 elements, where 2k < n <2k + 1. Here 2k + 1 is the smallest power of 2 larger than n).

  22. Example 4 Average case performance of Linear Search Algorithm 2i + 1 comparisons i = 1 : 3 i = 2 : 5 : i = n : 2n + 1 Average = (3 + 5 + 7 +n…+ 2n + 1)/2 = (2(1+2+n…+ n) + n)/n Since 1 + 2 + 3 +…+ n = n(n + 1)/2 So, the average = 2[n(n + 1)/2] + n = n + 1 + 1 = n + 2 = O(n)

  23. Polynomial worst-case complexity = tractable O(nb), b is an integer ≥ 1 But when b is large or C is very large  problem (takes too long time to solve) “Intractable” : problem that cannot be solved within Polynomial time.

  24. Big O estimates of time complexity cannot directly tell the actual amount of computer time used. • Knowing the constants C and k makes | f(n)| ≤ C |(g(n))|when n > k • The time each type of operations used is not equivalent. • Type of computers can also be different in terms of specification, and generation.

  25. Asymptotic Notations • Big O (Upper bound) • f(n) grow faster than g(n) • |f(n)|  C|g(n)| when n > k  f(n) = O(g(n)) • Big  (Lower bound) • f(n) grow slower than g(n) • |f(n)|  C|g(n)| when n > k  f(n) = (g(n)) • Big  (Tight bound) • f(n) and g(n) have the same rate of growth • C1|g(n)|  |f(n)|  C2|g(n)| when n > k  f(n) = (g(n)) • C and k called witness pair.

  26. The Integers & Division Definition 1: If a and b are integers with a ≠ 0, we say that a divides b if there is an integer c such that b = ac. When a divides b we say that a is a factor of b and that b is multiple of a. The notation a | b denotes that a divides b. We write a | b when a does not divide b. THEOREM 1: Let a, b and c be integers. Then 1. if a | b and a | c, then a | (b + c); 2. if a | b, then a | bc for all integers c; 3. if a | b and b | c, then a | c.

  27. Definition 2: A positive integer p greater than 1 is called prime if the only positive factors of p are 1 and p. A positive integer that is greater than 1 and is not prime is called composite. THEOREM 2: THE FUNDAMENTAL THEOREM OF ARITHMETIC Every positive integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of non-decreasing size.

  28. Example 1 The prime factorizations of 100, 641, 999, and 1024 are given by 100 = 2255 = 2252, 641 = 641, 999 = 33337 = 33  37, 1024 = 2222222222 = 210 THEOREM 3: If n is a composite integer, then n has a primedivisor less than or equal to

  29. THE DIVISION ALGORITHM THEOREM 6: THE DIVISION ALGORITHM Let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 ≤ r < d, such that a = dq + r

  30. Definition 3: In the equality given in the division algorithm, d is called the divisor, a is called the dividend, q is called the quotient, and r is called the remainder. Definition 4: Let a and b be integers, not both zero. The largest integer d such that d|a and d|b is called the greatest common divisorofaand b. The greatest common divisor of a and b is denoted by gcd(a, b).

  31. Definition 5: The integer a and b are relatively primeif their greatest common divisor is 1. Example: gcd (17, 22) = 1 17, 22 = relatively prime Definition 6: The integer a1, a2,…, an are pairwise relatively prime if gcd(ai, aj) = 1 whenever 1 ≤ i < j ≤ n.

  32. Example: 10, 17, 21 are pairwise relatively prime? 10, 19, 24 are pairwise relatively prime?

  33. Definition 7: The least common multiple of the positive integers a and b is the smallest positive that is divisible by both a and b. The least common multiple of a and b is denoted lcm(a, b). Example: What is the least common multiple of 233572 and 2433? Solution: We have lcm(233572, 2433) = 2max(3, 4)3max(5, 3)7max(2, 0) = 243572

  34. THEOREM 7: Let a and b be positive integers. Then ab = gcd(a, b)  lcm(a, b)

  35. MODULAR ARITHMATIC Let a be an integer and m be a positive integer. We denote “a mod m” the remainder when a is divided by m. Definition 8: If a and b are integers and m is a positive integer, then a is congruent to b modulo m, if m divides a - b. The notation a ≡ b (mod m)is used to indicate that a is congruent to b modulo m. If a and b arenot congruent modulo m, a ≡ b (mod m).

  36. Theorem 8: Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only ifa mod m = b mod m. Example: Determine whether 17 is congruent to 5 modulo 6 and whether 24 and 14 are congruent modulo 6. Solution: Since 6 divides 17 - 5 = 12, we see that 17 ≡ 5 (mod 6). However, since 24 - 14 = 10 is not divisible by 6, we see that 24 ≡ 14 (mod 6).

  37. Applications of Number Theory • Puzzle • Cryptography • Caesar’s Ciphering • RSA • Computer Graphics • Signal processing • Thermodynamics • Quantum Physics • etc. Chinese Remainder Theorem Modular Arithmetic

  38. Matrices • Matrix equality • Equals in number of rows and columns, and elements • Matrix Arithmetic • Addition, multiplication • Identity matrix • AIn = ImA = A • Transpose and Inverse matrix • Boolean product

  39. Administrivia • Homework due today • 1.6-1.8 • Homework 3 is out today • Study by yourself • Prolog and Python • Programming Assignments will be coming soon!!

  40. Proof Review: Basic Proof Methods

  41. Nature & Importance of Proofs • In mathematics, a proof is: • A sequence of statements that form an argument. • Must be correct (well-reasoned, logically valid) and complete (clear, detailed) that rigorously & undeniably establishes the truth of a mathematical statement. • Why must the argument be correct & complete? • Correctness prevents us from fooling ourselves. • Completeness allows anyone to verify the result.

  42. Rules of Inference • Rules of inference are patterns of logically valid deductions from hypotheses to conclusions. • We will review “inference rules” (i.e., correct & fallacious), and “proof methods”.

  43. Visualization of Proofs A proof A Particular Theory Rules Of Inference … The Axiomsof the Theory Various Theorems

  44. Inference Rules - General Form • Inference Rule – • Pattern establishing that if we know that a set of hypotheses are all true, then a certain related conclusion statement is true. Hypothesis 1 Hypothesis 2 …  conclusion “” means “therefore”

  45. Inference Rules & Implications Each logical inference rule corresponds to an implication that is a tautology. Hypothesis 1 Inference rule Hypothesis 2 …  conclusion Corresponding tautology: ((Hypoth. 1)  (Hypoth. 2)  …)  conclusion

  46. Formal Proofs • A formal proof of a conclusion C, given premises p1, p2,…,pnconsists of a sequence ofsteps, each of which applies some inference rule to premises or to previously-proven statements (as hypotheses) to yield a new true statement (the conclusion). • A proof demonstrates that if the premises are true, then the conclusion is true (i.e., valid argument).

  47. Common Fallacies • A fallacy is an inference rule or other proof method that is not logically valid. • May yield a false conclusion! • Fallacy of affirming the conclusion: • “pq is true, and q is true, so p must be true.” (No, because FT is true.) • Fallacy of denying the hypothesis: • “pq is true, and p is false, so q must be false.” (No, again because FT is true.)

  48. Common Fallacies - Examples “If you do every problem in this book, then you will learn discrete mathematics. You learned discrete mathematics.” p: “You did every problem in this book” q: “You learned discrete mathematics” • Fallacy of affirming the conclusion: pq and q does not imply p • Fallacy of denying the hypothesis: pq and  p does not imply  q

  49. Inference Rules for Quantifiers • xP(x)P(o) (substitute any object o) • P(g) (for g a general element of u.d.)xP(x) • xP(x)P(c) (substitute a newconstantc) • P(o) (substitute any extant object o) xP(x) Universal instantiation Universal generalization Existential instantiation Existential generalization

More Related