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Chapter 4

Chapter 4 . Properties of the Integers: Mathematical Induction. Introduction. Mathematical induction 數學歸納法 is a technique exhibited by the subset of positive integers.

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Chapter 4

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  1. Chapter 4 Properties of the Integers: Mathematical Induction

  2. Introduction • Mathematical induction 數學歸納法is a technique exhibited by the subset of positive integers. • Four sets of numbers that are very important in the study of discrete mathematics and combinatorics are introduced – namely, the harmonic numbers調和數列, the Fibonacci numbers費伯納西數列, the Lucas numbers, and the Eulerian numbers. • When x, yZ, we know that x + y, xy, x–yZ. Thus we say that the set Z is closed under (the binary operations of) addition, multiplication, and subtraction. • Note that 2, 3 Z but that the rational number 2/3 is not a member of Z. So the set Z of all integers is not closed under the binary operation of nonzero division. • A special concentration on primes (Z+).

  3. 4.1 The Well-Ordering Principle: Mathematical Induction • The Well-Ordering Principle (of positive integers):Every nonempty subset of Z+ contains a smallest element. (often saying that Z+ is well ordered) • The set Z+ is different from the sets Q+ and R+ in that every nonempty subset X of Z+ contains an integer a such that ax, for all x X.

  4. Theorem 4.1: Finite Induction Principle, or Principle of Mathematical Induction. • Let S(n) denote an open mathematical statement (or set of such open statements) the involves one or more occurrences of the variable n, which represents a positive integer. • If S(1) is true (referred to as the basic step); and • If whenever S(k) is true (for some particular, but arbitrarily chosen, k Z+), then S(k +1) is true (called the inductive step); • then S(n) is true for all n Z+.

  5. Mathematical Induction • Proof: see page 164. • The choice of 1 in the first condition of Theorem 4.1 is not mandatory. If the true of S(n0) is for our basic step (a different starting point!), we can rewrite the Finite Induction Principle, using quantifiers, as • [S(n0)  [kn0 [S(k) S(k + 1)]]] nn0S(n).

  6. Example 4.1: page 166.

  7. Example 4.3: page 167.

  8. Example 4.4: page 168. • (This example indicates the effect of the wrong starting point –n 7 or n 137.)

  9. Example 4.7 (the harmonic numbers H1, H2, …): page 171.

  10. Example 4.9: page 172. • (Program verification by using mathematical induction!)

  11. Theorem 4.2: Finite Induction Principle–Alternative Form. • Let S(n) denote an open mathematical statement (or set of such open statements) that involves one or more occurrences of the variable n, which represents a positive integer. Also let n0, n1 Z+ with n0n1. • If S(n0), S(n0 + 1), S(n0 + 2), …, S(n1– 1), and S(n1) are true; and • If whether S(n0), S(n0 + 1), …, S(k– 1), and S(k) are true for some (particular but arbitrarily chosen) k Z+, where kn1, then the statement S(k + 1) is also true; • then S(n) is true for all nn0.

  12. Example 4.11: page 175.

  13. Example 4.12: page 175.

  14. 4.2 Recursive Definitions • The terms of some integer sequences may have explicit formula to define, for example, b0 = 2  0 = 0, b1 = 2  1 = 2, b2 = 2  2 = 4, b3 = 2  3 = 6, …, where bn = 2 n for all nN. • Other sequences may not have explicit formula, for example, a0 = 1, a1 = 2, a2 = 3, and an = an-1 + a= + an-3, for all nN where n 3. • The way we define for a0, a1, a2, …, an, … is called a recursive definition. The assignments a0 = 1, a1 = 2, a2 = 3 provide a base for the recursion. The equation an = an-1 + an-2 + an-3, for all nN where n 3, provides the recursive process; it indicates how to obtain new entries in the sequence from those prior results we already know.

  15. Example 4.13 (Generalized Associative Law for ),

  16. Example 4.14 (Generalized Associative Law for ),

  17. Example 4.15 (Generalized DeMorgan Law):

  18. Example 4.16 (the Fibonacci numbers): see page 183; • the recursive definition of the Fibonacci numbers can be used to establish many of the interesting properties that these numbers exhibit. One of these properties is as follows: • nZ+

  19. Proof: see page 184.

  20. Example 4.17 (the Lucas numbers): see page 184~185. • (One of the interrelations between the Fibonacci and Lucas numbers is illustrated in the fact that nZ+ .)

  21. (Note: for a fixed mZ+, .) Example 4.18 (the Eulerian numbers): see page 185~186.

  22. 4.3 The Division Algorithm: Prime Numbers • Definition 4.1: If a, bZ and b ≠ 0, we say that b divides a, and we write b|a, if there is an integer n such that a = bn. When this occurs we say that b is a divisor of a, or a is a multiple of b.

  23. Theorem 4.3: For all a, b, cZ • a) 1|a and a|0. b) [(a|b)  (b|a)] a = b. • c) [(a|b)  (b|c)] a|c. d) a|ba|bx for all xZ. • e) If x = y + z, for some x, y, zZ, and a divides two of the three • integers x, y, and z, then a divides the remaining integer. • f) [(a|b)  (a|c)] a|(bx + cy), for all x, yZ. (The expression bx + cy is called a linear combination ofb, c.) • g) For 1in, let ciZ. If a divides each ci, then a|(c1x1 + c2x2 + …+ cnxn) , where xiZ for all 1in.

  24. Example 4.20 and 21: page 190.

  25. Lemma 4.1: • If n Z+ and n is composite, then there is a prime p such p | n.

  26. Theorem 4.4: (Euclid) There are infinitely many primes.

  27. Theorem 4.5:The Division Algorithm. • If a, b Z, with b > 0, then there exist unique q, r Z with a = qb + r, 0 r < b.

  28. Example 4.22: page 192.

  29. Example 4.24, page 194

  30. Example 4.25

  31. Example 4.26, 4.27: page 196~198. • (Note on one’s and two’s complements!)

  32. 4.4 The Greatest Common Divisor: The Euclidean Algorithm

  33. Definition 4.2: • For a, bZ, a positive integer c is said to be a common divisor of a and b if c|a and c|b.

  34. Example 4.29: page 200.

  35. Definition 4.3: • Let a, bZ, where either a 0, or b 0. Then cZ+ is called a greatest common divisor of a, b if c|a and c|b (that is, c is a common divisor of a, b), and • for any common divisor d of a and b, we have d|c.

  36. Theorem 4.6: • For all a, bZ+, there exists a unique cZ+ that is the greatest common divisor of a, b (abbreviated as gcd(a,b)). • Proof: See page 201.

  37. From Theorem 4.6 we see that not only does gcd(a,b) exist but that gcd(a,b) is also the smallest positive integer we can write as a linear combination of a and b. • Integers a and b are called relatively prime when gcd(a,b) = 1 – that is, when there exist x, yZ with ax + by = 1.

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