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
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No, the diagram above does not define a function. The element a is not sent to a unique element of Y; there are two arrows coming out of a, one pointing to 1 and the other pointing to 2.
Answer the following questions for the arrow diagram below:
1
2
3
w x y z X
Y

2. Does the diagram above represent a function f: X Y? Explain why or why not
Yes, the diagram above represents a function. All elements of X have an arrow coming out of it also, no element of X has two arrow coming out of it that point to two different elements of Y. Functions also may have elements of the co-domain with no matching elements in the domain (z).

3. If so, what is the Domain?
Domain = {1, 2, 3} 4. If so, what is the Co-Domain?
Co-Domain = {w, x, y, z} 5. If so, what is the Range?
Range = {w, x, y} 6. If so, what is the value of f (3)?
F (3) = y 7. If so, what is the Inverse Image of x?
Inverse image of x = {2} 8. If so, is the function one-to-one? Explain why or why not
Yes, the function above is a one-to-one function. There are distinct points in the domain that map to distinct points in the co-domain. 9. If so, is the function onto? Explain why or why not
No, the function is not onto. This is because z ≠ F(x) for any x in x. 10. Represent the arrow diagram as a set of ordered pairs
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{1,w},{2,x},{3,y}
Answer the following questions for the arrow diagram below:

11. Does the diagram above represent a function f: X Y? Explain why or why not
Yes, the diagram above does represent a function. First, every element of X has an arrow coming out of it. Second, No element of X has two arrows coming out of it that point to two different elements of Y. Functions can have more than one element of the domain map to the same element of the co-domain. 12. If so, is the function one-to-one? Explain why or why not
No, the function doesn’t take distinct points in the domain to distinct points in the co-domain; a and b points point to the same co-domain of 1. At least two points of the domain are taken to the same point in the co-domain. 13. If so, is the function onto? Explain why or why not
No, this is not an onto function. The elements in the co-domain (2, 3) do not match any element in X or
2 ≠f(x) and 3 ≠ f(x)
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Answer the following questions for the arrow diagram below:

14. Does the diagram above represent a function f: X Y? Explain why or why not
Yes, the diagram above does represent a function. First, every element in X has an arrow coming out of it. Second, no element in x has two arrows coming out of it that point to two different elements of Y. 15. If so, is the function one-to-one? Explain why or why not
Yes, the function is a one-to-one function. This is because no two different elements of X are sent to the same element of Y, each is distinct. Distinct points in the domain are mapped to distinct points in the co-domain. 16. If so, is the function onto? Explain why or why not
Yes, the function is an onto function. This is because each element of Y equals f(x) for some x in x.

17. If so, is the function a one-to-one correspondence? Explain why or why not
Yes, the function is a one-to-one correspondence. A function that is one-to-one and onto sets up a pairing between the elements of X and the elements of Y that matches. Given any element x in X, there is a unique corresponding element y = f(x). This defines the diagram above is an one-to-one correspondence. 18. If it is a one-to-one correspondence, give its inverse as a set of ordered pairs
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{(4,d) (3,c)(2,b)(1,a)}
Answer the following questions for the arrow diagram below:

19. Does the diagram above represent a function f: X Y? Explain why or why not
Yes, the diagram above does represent a function. Every element of X has an arrow coming out of it. Second, No element of X has two arrows coming out of it that point to two different elements of Y. 20. If so, is the function one-to-one? Explain why or why not
No, the function is not one-to-one. There are two distinct elements of X (a, b) that are sent to the same element of Y (1). 21. If so, is the function onto? Explain why or why not
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Yes, this is an onto function. Each element in y in Y equals F(x) for at least one x in X. 22. 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 53, 38, 42, 22, and 76 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:
Hash function definition: h(x) =?

Address | Contents | 0 | | 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | |

23. Prove that F: R R where F(x) = 5x+7 is one-to-one.
Let x1 and x2 belongs to R
And x1=x2
=> 5*x1= 5*x2
=>5*x1+7=5*x2+ 7 (Divide both sides by 5) (Subtracting 7 from both sides)
=> F(x1) =f(x2) (Therefore)
This means that whenever x1=x2, then f(x1) =f(x2),
Therefore, one-one

24. Discuss the role of inverse functions in simple cipher encryption (also indicate whether simple ciphers are strong enough for modern commercial encryption needs)

For the remaining problems, calculate the log and justify why your calculation is correct by using the definition of log as in the online Course Content: 25. Log10(100,000) = 5

Because: 105 = 100,000 26. Log2(128) = 7

Because: 27 = 128 27. Log3(243) = 5

Because: 35 = 243 28. Log10(1/1,000) = -3

Because 10-3 = .001