ДискретнаяМатематика / Student Solutions Manual / chapter 4

1. (a) Q The function F (x) = ax + b where a; b 2 Q is onto if for any y0 2 Q there is an x0 such that y0 = ax0 + b: Solving this equation for x0 is always possible provided a 6= 0:

1. (b) Z: The function F (x) = ax + b where a; b 2 Z is onto if for any y0 2 Z there is an x0 such that y0 = ax0 + b: Solving this equation for x0 gives x0 = (y0 b)=a: This equation can be solved in Z for all y0 2 Z provided a = 1 or a = 1:

1. (c) N: The function F (x) = ax + b where a; b 2 N is onto if for any y0 2 N there is an x0 such that y0 = ax0 + b: Solving this equation for x0 gives x0 = (y0 b)=a: For this equation to be solved in N for all y0 2 Z not only does a have to be 1, but also b must be zero. If b is a nonzero natural number then no natural number between 0 and b would be the image of any x 2 N.

3.Show that the set f3, 6, 9, 12, . . . g is countable.

3. The function F : N ! f3; 6; 9; 12; : : :g, where F (n) = 3(n + 1) is a bijection.

5.(a) The lattice points in the plane R2 are the points (x; y) where x and y are both integers. Prove that there are only countably many lattice points in R2:

(b) The lattice points in three-dimensional space R3 are the points (x; y; z) where x; y, and z are all integers. Prove that there are only countably many lattice points in R3:

5. (a) Use the diagram that proves that the rationals are countable but interpret the point as ordered pairs rather than quotients.

5.(b) Consider the product N X where X consists of the ordered pairs as they were enumerated in part (a). Now look at this product as a two dimensional plane and traverse the points of the form (i; (x; y)) using the grid used to show the rationals are countable. This will enumerate all the 3-dimensional lattice points and prove this is a countable set.

7.Let S be a set of six positive integers who's maximum is at most 14. Show that the sums of the elements in all the nonempty subsets of S cannot all be distinct.

7.For any nonempty subset A of S, the sum of the elements in A, denoted as SA; satis es

1 SA 9 + 10 + 11 + 12 + 13 + 14 = 69

There are 26 1 = 63 nonempty subsets of S. There are too many pigeonholes. So{

Consider the nonempty subsets A where j A j 5: Then

1 SA 10 + 11 + 12 + 13 + 14 = 60

Since there are 62 nonempty subsets of S with A 5, the elements of at least two of these subsets must be the same.

9.(Uses First Order Logic:) This exercise concerns a result needed for the de nition of subsequence. We took it as obvious there, but it does need proof. Let X N.

Part of the job here is to use as simple set theory as possible to de ne the

functions. You may assume that (i) sets ;, X, N, and N X exist; (ii) for every i; j 2 N, (i; j) exists; (iii) for any x, fxg exists; (iv) for any sets x; y, x[y and x y exist; and (v) the collection of elements of any set satisfying some formula of First Order Logic (with quanti ers ranging over numbers and sets) exists. You are to use just this much set theory to show that the function de ned by recursion exists. The problem is that induction may naturally be used to prove that arbitrarily large, nite sets of ordered pairs exist, but it does not allow one to conclude that a particular in nite set of ordered pairs exists.

Prove the following:

(a)Suppose X is nite. Let n = j X j. There is at most one increasing function from f0; 1; : : : ; n 1g onto X. (Hint: Suppose there are two, say, F and G. Prove by induction on i < n that F (i) = G(i):)

(b)If X is in nite, there is at most one increasing function from N onto

X.

9. (a) Solution Required

9. (b) Solution Required

Using Discrete Mathematics in Computer Science

1.Let F be the function that maps strings of characters and blank spaces onto strings of characters by removing all blank spaces and all vowels. For example, F (\dog cat") = \dgct." Let G be the function that maps strings of characters onto integers such that the value of a string is simply the number of characters in the string. What is F (\george washington")? What is G(\george washington")? What is G F (\george washington")? The function F is a simple example of a compression technique.

1. F (\george washington") =

grgwshngtn." G(\george washington") = 17. G F (\george washington") = 10.