# (a) How many bit strings of length 8 are there? Explain. (b) How many bit strings of length 8 are there which begin with a 0 and end with a 0?

Discrete Structures I

1. (15 points)

(a) How many bit strings of length 8 are there? **Explain.**

(b) How many bit strings of length 8 are there which begin with a 0 **and** end with a 0? **Explain.**

(c) How many bit strings of length 8 are there which contain at most 3 ones? **Be careful with this one. Explain. (See your notes, week 10.)**

(d) How many bit strings of length 8 are palindromes?

3. (10 points)

(a) Text, page 405, number 2. **Explain. **

Show that if there are 30 students in a class, then at least two have last names that begin with the same letter.

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(b) Text, page 406, number 36. **Explain**

A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.

3. (10 points).

Text, page 414, number 30. **Explain. **

Seven women and nine men are on the faculty in the mathematics department at a school.

a) How many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee?

b) How many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee?

4. (10 points)

(a) Use the Binomial Theorem to write the expansion of (x + y)^{ 6}?

(b) Write the coefficient of the term x^{2}y^{4}z^{5} expansion of (x + y + z)^{ 11}. See example 12 in your notes of week 10

5. Study pages 1-4 of the notes for week 11. Let A = {a, b, c, d}, and let R be the relation defined on A by the following matrix:

M_{R} =

(a) (10 pts.) Describe R by listing the **ordered pairs** in R and draw the **digraph** of this relation.

(b) (15 pts.) (Note this is similar to exercise 7 page 630. Which of the properties: reflexive, antisymmetric and transitive are true for the given relation? Begin your discussion by defining each term **in general **first and then how the definition relates to this specific example.

(c) (5 pts.) Is this relation a partial order? **Explain**. If this relation is a partial order, draw its Hasse diagram.

5. (10 points) Use the Hasse diagram of number 26 page 631

(a) List the ordered pairs that belong to the relation. Keep in mind that a Hasse diagram is a graph of a partial ordering relation so it satisfies the three properties listed in number 5 part (b).

(b) Find the (Boolean) matrix of the relation.

7. (15 points) Before you do this problem study the example at the end of the exam, as well as the notes in weeks 12 and 13.

Assume the Boolean matrix below is M_{R} and that M_{R} represents the relation R where R represents the connecting flights that an airline has between 4 cities: a, b, c, and d.

The 1 in row **a** column **b** means there is a flight from city **a** (Manchester) to city **b** (Boston). The 1 in row x column x means that there are planes in airport x. In general, there is a 1 in row x column y iff there is a connecting flight between (from) city x and (to) city y That is, the rows of the matrix represent the cities of the origins of the flights and the columns represent the destination cities.

Let M_{R} =

(i) Let a stand for the airport in the city of Manchester, let b stand for the airport in Boston, c stand for the Chicago airport, d for the airport in the city of Denver. Is their a flight from Denver to Chicago? **Explain.**

(ii) Compute and interpret the Boolean products: M_{R}^{ 2}, and M_{R}^{ 3}. (Remember to use Boolean arithmetic). What do these Boolean products give you? That is, explain what the Boolean entries in the matrices M_{R}^{ 2} and M_{R}^{ 3} mean.

(iii) Now call the given matrix A and compute A^{2} and A^{3} , using **regular**, not Boolean arithmetic. What do these products give you?

(v) What does M_{R} + M^{2}_{R} + M^{3}_{R} + M^{4}_{R} give you? Note, this is Theorem 3 page 602where the text’s symbol, v, for Boolean addition is replaced by +, another symbol for Boolean addition .

**Bonus questions**

- Make up a question using the idea of question 7 (Something you do at work, home, something you’re interested in.) See 3 below for an example you can do.
- In your notes of week 12 Project Evaluation and Review Technique (PERT) is explained. Make up a meaningful example illustrating this process. Explain all details. Draw the graphs involved. You should use a reasonable number of tasks.

(a) Determine the minimum time needed to complete your set of tasks.

(b) Determine the Critical Path for your example. Explain what this gives you.

(c) Comment on your example. Which task(s) cause a delay in the project? How would you fix the problem? Can you obtain more information out of your example?

- The Network analysis problem.
- Read Program Evaluation and Review Technique (PERT) in your week 14 notes. Note the examples given here are a little “ambitious”. Make up an example using PERT and find and explain the critical path of your example. Google “critical path analysis”, for more ideas.
- Pg 406 #38

**If #7 is not clear here is an example which may help you to understand #7**

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Let D= days of the week {M, T, W, R, F},

E = {Brian (B), Jim (J), Karen (K)} be the employees of a tutoring center at a University and let

U = {Courses the tutoring center needs tutors for}

= {Calculus I (I), Calculus II (II), Calculus III (III), Computers I (C_{1}), Computers II (C_{2}), Precalculus (P)}.

We define the relation R from *D *into E by *d *R e, if employee e is scheduled to work on day *d. *We also define S from E into U by e r u, if employee e is capable of tutoring students in course u.

For example, the matrix M_{R} indicates that on R (Thursday) that J (Jim) is available to tutor but Brian and Karen are not.

Assume M_{R} = and M_{S} =

(a) Interpret the above matrices with respect to the above relations.

(b) Compute, (use Boolean arithmetic) and use the matrix to determine which courses will have tutors available on which days.

(c) Multiply the above matrices using **regular** arithmetic. Can you interpret this result?

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