# Module Six Homework

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**Section 5.1 Homework**

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1) Let A = {1, 2, 3, 4, 5} and B = {MA, NH, NV, TX, AK, ME}.

- Define a relation R from A to B that is a function and contains at least 4 ordered pairs.
- What is the domain of this function?
- What is the range of this function?

This problem is similar to example 2 and problems 5.1.1 and 5.1.2.

2) Define functions f: ℝ → ℝ and g: ℝ → ℝ by f(a) = 5a – 3 and g(b) = 4 – 2b. Find the following:

- (f○g)(0)
- (g○f)(1)
- (f○g)(x)
- (g○f)(x)

This problem is similar to example 9 and problems 5.1.9 and 5.1.10.

3) Let A = {2, 3, 5, 7, 11, 13, 17, 23} and B = {a, e, i, o, u, y}. Using at least 5 ordered pairs, define the following:

- A function from A to B that is one-to-one.
- A function from A to B that is not one-to-one.
- A function from A to B that is onto.
- A function from A to B that is not onto.
- A function from B to A that acts as the inverse of the function you created in part a) of this problem.

This problem is similar to example 12 and problems 5.1.11 and 5.1.12.

4) The function f: ℝ → ℝ defined by f(x) = x³ is onto because for any real number , we have that is a real number and . Consider the same function defined on the integers g: ℤ → ℤ by g(n) = n³. Explain why g is not onto ℤ and give one integer that g cannot output. This problem is similar to examples 10 and 12.

5) Let A = {x| x is a nation}

B = {Asia, Africa, North America, South America, Antarctica, Europe, Australia}

Let f:A→B be the function that outputs the continent to which a nation belongs. For example, f(Iceland) = Europe and f(Greenland) = North America. Explain why f is not a one-to-one function and give an example to prove it. This problem is similar to example 16 and problems 5.1.11b and 5.1.12b.

**Section 5.2**

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1) Suppose a health insurance company identifies each member with a 7-digit account number. Define the hashing function *h* which takes the first 3 digits of an account number as 1 number and the last 4 digits as another number; adds them, and then applies the mod-41 function.

- How many linked lists does this create?
- Compute h(4686158)
- Compute h(9813284)

This problem is similar to example 10 and problems 5.2.24–5.2.26.

2) Compute the check digit *c* for the following ISBNs.

- 031676948-
*c* - 140123517-
*c*

* *

This problem is similar to problems 5.2.49 and 5.2.50.

**Section 5.3 Homework**

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1) The picture below shows the graph of r(x) in red and the graph of b(x) in blue. Does this graph show that r is *O*(b) or that b is *O*(r)? Explain.

**Section 6.1 Homework**

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1) Define a relation R on the set of real numbers by (x,y) R if and only if

x – y = 0. Determine if the relation R is a partial order. If it is not a partial order, explain which property or properties it fails to have. This problem is similar to example 4 and problems 6.1.1 and 6.1.2.

2) Determine the ordered pairs in the relation determined by the Hasse diagram on the set A = {a, b, c, d, e}. Create the matrix representation of this poset. This problem is similar to example 11 and problems 6.1.11 and 6.1.12.

3) Define U = {1, 2, 3, 4, 5}. Consider the following subsets of U:

P = {2, 3, 5}, O = {1, 3, 5}, E = {2, 4}, S = {3}

- Create the Hasse diagram using Í as the partial order on sets E, O, P, S, U, and Æ. This problem is similar to example 12 and problem 6.1.21–6.1.24

- b) Is this a linear order? Explain.

4) If represents the lexicographic order, which of the following is/are true?

- (5,12) (5,4)
- (5,4) (5,12)
- (5,4) and (5,12) are not comparable because we need the first number to be smaller in one of the pairs.

This problem is similar to problems 6.1.19 and 6.1.20.

**Section 6.2 Homework**

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1) Let B = {1, 2, 3, 6, 12, 18} and R be defined by xRy if and only if x|y.

- Determine all minimal and all maximal elements of the poset.
- Find all least and greatest elements of the poset.

This problem is similar to examples 1–3 and 5–7 and problems 6.2.8 and 6.2.16.