Discrete Structures

# HW #9

 Last Update: 27 March 2009 Note: or material is highlighted

Reminder: Each HW problem solution should consist of:

• a restatement of the entire problem (you may copy it word for word),
• followed by a complete solution with all intermediate steps shown.

All exercises are from §2.3 (functions) and §2.4 (sequences and summations).

1. (3 points each; total = 9 points)

p. 146: 12a, b, c

• You are given some functions and asked to determine which are 1-1.
• For full credit, you must justify your answers.

2. (3 points each; total = 9 points)

pp. 146-147: 14a, c, e

• You are given some functions and asked to determine which are onto.
• For full credit, you must justify your answers.

3. (3 points each; total = 6 points)

Let R = {x | x is a real number}.
Let f, g : RR such that (∀xR)[f(x) = x3g(x) = 2x]

Compute:

1. (f o g)(x)
2. (g o f)(x)

4. (6 points)

p. 148: 68.

• You are asked to prove that a certain kind of function is 1-1 iff it is onto.
• Hint: Use the definitions of "1-1" and of "onto" together with the fact that:
if |A| = |B|, then:
if a function from one of these to the other is not 1-1 or not onto, then |A| ≠ |B|.

5. (3 points)

p. 149: 74a

• You are asked to prove that a partial function (i.e., a function that is not defined on some elements of its domain) can be "extended" to a total function by assigning an arbitrary image to each element for which it is not defined.
• Hint: All you have to do is show that f * satisfies the definition of a function.

6. (3 points)

p. 161: 4a

• You are asked to compute the first 4 terms of a sequence.

7. (3 points)

List the first 10 terms of the sequence {an} whose first two terms are a0 = –3 and a1 = 2, and which is such that each succeeding term is the sum of the two previous terms.

• I.e., an+2 = an + an+1

8. (3 points)

p. 161: 16a

• You are asked to compute the value of a summation (a.k.a., a "series"), i.e., to compute the sum of the terms of a sequence.
• Hint: Compute the terms of the sequence, and then add them up!

Total points = 42.

```A       41 - 42
A-      38 - 40
B+      36 - 37
B       34 - 35
B-      31 - 33
C+      29 - 30
C       24 - 28
C-      20 - 23
D+      15 - 19
D        8 - 14
F        0 -  7
```

 DUE AT BEGINNING OF LECTURE, FRIDAY, APRIL 3

Copyright © 2009 by William J. Rapaport (rapaport@cse.buffalo.edu)
http://www.cse.buffalo.edu/~rapaport/191/S09/hw09.html-20090322