0.5 Discrete structures recursion  (Page 8/8)

Proof: Assume that for all positive integers k, n>k>1, k can be written as the product of prime numbers.

We are going to prove that n can be written as the product of prime numbers.

Since n is an integer, it is either a prime number or not a prime number. If n is a prime number, then it is the product of 1, which is a prime number, and itself. Therefore the statement holds true.

If n is not a prime number, then it is a product of two positive integers, say p and q. Since both p and q are smaller than n, by the induction hypothesis they can be written as the product of prime numbers (Note that this is not possible, or at least very hard, if the First Principle is being used). Hence n can also be written as the product of prime numbers.

Questions and exercises

1. Indicate which of the following statements are correct and which are not.

a. The number 23 can be generated for EI in Example 3 in Section Recursive Definition.

b. Basis and Inductive Clauses are sufficiency for membership for the set.

c. The set {4} can replace the basis for NE of Example 2 in Section Recursive Definition.

d. If empty set is the basis of S in Example 4 in Section Recursive Definition, then the string ab is in S.

2. Indicate which of the following statements are correct and which are not.

a. Algorithm 2 in Section Recursive Algorithm produces 0, 2, 6 and 8 when computing the third power of 2.

b. Recursive algorithms are good because they run more efficiently than iterative ones.

c. In Algorithm 3 in Section Recursive Algorithm, x is first compared with the key at the middle of L.

d. If the input to Algorithm 1 in Section Recursive Algorithm is not a natural number, then 0 is returned.

3. Look at the Section Mathematics Induction; indicate which of the following statements are correct and which are not.

a. In the Inductive Step, P(n) is proven assuming that P holds for the parent of n.

b. In the Inductive Step, since we assume P(k) for an arbitrary k, P(k+1) holds.

c. The Induction Hypothesis does NOT assume P(k) for all k.

d. In the Induction, since k is arbitrary, we can prove P(6) assuming P(5) holds.

e. The Basis Step proves the statement for the elements of the basis.

4. Look at the Section Mathematics Induction; indicate which of the following statements are correct and which are not.

a. In the Second Principle, P(k) is assumed true for one arbitrary value of k.

b. The Second Principle does not make a proof any easier.

c. The Basis Step of the First Principle is implicitly proven by the Second Principle.

d. The Second Principle can be applied when n starts at some integer larger than 0.

e. The Second Principle gives you more assumptions to use, making a proof easier.

5. Let A i = { 1, 2, 3, ..., i } for i = 1, 2, 3, ... . Find ${}_{i=1}^n{A}_i$

6. Let A i = { i, i+1, i+2, ... } for i = 1, 2, 3, ... . Find ${\mathrm{intersect}}_{i=1}^n{A}_i$

7. Give a recursive definition of the set of positive integers that are multiples of 5.

8. Give a recursive definition of

a. the set of even integers.

b. the set of positive integers congruent to 2 modulo 3.

c. the set of positive integers not divisible by 5.

9. When does a string belong to the set A of bit strings (i.e. strings of 0's and 1's) defined recursively by

Basis Clause: ∅ ∈ A

Inductive Clause: 0 x 1 ∈ A   if x ∈ A

where ∅ is the empty string (An empty string is a string with no symbols in it.)

Extremal Clause: Nothing is in A unless it is obtained from the Basis and Inductive Clauses.

10. Find   f(1) ,  f(2) ,  and   f(3) ,   if   f(n) is defined recursively by   f(0) = 2   and   for n = 0, 1, 2, ...

a. f(n + 1) = f(n) + 2.

b. f(n + 1) = 3f(n).

c. f(n + 1) = 2 f(n) .

11. Find   f(2) ,   f(3) ,  and   f(4) ,   if   f(n) is defined recursively by   f(0) = 1 ,   f(1) = -2   and  for n= 1, 2,...

a. f(n + 1) = f(n) + 3f(n - 1).

b. f(n + 1) = f(n) 2 f(n - 1).

12. Let F be the function such that F(n) is the sum of the first n positive integers.  Give a recursive definition of F(n) .

13. Give a recursive algorithm for computing nx whenever n is a positive integer and x is an integer.

14. Give a recursive algorithm for finding the sum of the first n odd positive integers.

15. Use mathematical induction to prove that 3 + 3 * 5 + 3 * 52+ ... + 3 * 5n = 3 (5n+1 - 1)/4 whenever n is a nonnegative integer.

16. Prove that 12 + 32 + 52+ ... + (2 n + 1)2 = ( n + 1)(2 n + 1)(2 n + 3)/3 whenever n is a nonnegative integer.

17. Show that 2n> n 2whenever n is an integer greater than 4.

18. Show that any postage that is a positive integer number of cents greater than 7 cents can be formed using just 3-cent stamps and 5-cent stamps.

19. Use mathematical induction to show that 5 divides n 5- n whenever n is a nonnegative integer.

20. Use mathematical induction to prove that if A 1, A 2, ... A n are subsets of a universal set U , then ${\overline{{}_{i=1}^{n}A}}_i$   =  ${\mathrm{intersect}}_{i=1}^n{\overline{A}}_i$

21. Find a formula for

1/2 + 1/4 + 1/8 + ... + 1/2n

by examining the values of this expression for small values of n . Use mathematical induction to prove your result.

22. Show that if   a 1, a 2, ..., a n   are n distinct real numbers, exactly n - 1 multiplications are used to compute the product of these n numbers no matter how parentheses are inserted into their product.  ( Hint : Use the second principle of mathematical induction and consider the last multiplication).