Answer the following questions for the arrow diagram below:
1. Does the diagram above represent a function f: X → Y? Explain why or why not
2. If so, is the function one-to-one? Explain why or why not
3. If so, is the function onto? Explain why or why not
4. If this is a function, does it have an inverse? Explain why or why not
Answer the following questions for the set of ordered pairs below:
{ (a,1), (b,2), (b,3), (c,4) }
5. Does the set of ordered pairs above represent a function f: X → Y? Explain why or why not
6. Give a formal proof that the Successor Function, f(k) = k + 1 for all integers k, is a one-to-one correspondence.:
Answer the following questions for the arrow diagram below:
7. Does the diagram above represent a function f: X → Y? Explain why or why not
8. If so, what is the Domain?
9. If so, what is the Co-Domain?
10. If so, what is the Range?
11. If so, what is the value of f(2)?
12. If so, what is the Inverse Image of d?
13. If so, is the function one-to-one? Explain why or why not
14. If so, is the function onto? Explain why or why not
15. Represent the arrow diagram as a set of ordered pairs
Answer the following questions for the ordered pairs below:
{ (a,1), (b,2), (c,1), (d,3) }
16. Do the ordered pairs above represent a function f: X → Y? Explain why or why not
17. If so, is the function one-to-one? Explain why or why not
For the following problems, calculate the log AND justify why your calculation is correct by using the definition of log as in the online Course Content:
18. Log5(125) =
19. Log2(256) =
20. Log10(1/100,000) =
21. Using the Division Remainder Method, give a concrete example of a hash function for a storage array with 10 storage locations addressed 0 through 9. Call your hash function h, define your function mathematically as h(x) = x mod 10, then use it to show how records with key fields of 36, 39, 25, 45, 27, 35, and 53 (in that order) would be placed into storage using the diagram below. Don't worry about the Packing Factor or using a prime divisor. Use Linear Probing to handle any collisions:
Address Contents
0
1
2
3
4
5
6
7
8
9
22. Prove that F: R → R where F(x) = 4x - 7 is one-to-one
23. Prove that F: R → R (R is the set of all Real Numbers) where F(x) = 2x+3 is onto
24. a) Discuss the role of inverse functions in simple cipher encryption, and b) indicate whether simple ciphers are strong enough for modern commercial encryption needs
25. Draw an Arrow Diagram for the function shown in the following Input/Output Table (you may use Word's drawing tools or copy and paste from your favorite drawing tool or draw by hand and scan into your Quiz document):
INPUT OUTPUT
P Q S
1 1 0
1 0 1
0 1 1
0 0 0