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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. (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: 1
M R = 1 0 1 0 0 1 1 0 0 0 1 1 1 1 0 1 (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. 2
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