missed: Points _____ Total score: _____/100 points
Student's Name: _______________________________________
East Tennessee State University Department of Computer and Information Sciences CSCI 2710 (Tarnoff) Discrete Structures TEST 2 for Fall Semester, 2004 Read this before starting! This test is closed book and closed notes You may NOT use a calculator All answers must have a box drawn around them. This is to aid the grader (who might not be me!) Failure to do so might result in no credit for answer. If you perform work on the back of a page in this test, indicate that you have done so in case the need arises for partial credit to be determined. Statement regarding academic misconduct from Section 5.7 of the East Tennessee State University Faculty Handbook, June 1, 2001:
"Academic misconduct will be subject to disciplinary action. Any act of dishonesty in academic work constitutes academic misconduct. This includes plagiarism, the changing of falsifying of any academic documents or materials, cheating, and the giving or receiving of unauthorized aid in tests, examinations, or other assigned school work. Penalties for academic misconduct will vary with the seriousness of the offense and may include, but are not limited to: a grade of 'F' on the work in question, a grade of 'F' of the course, reprimand, probation, suspension, and expulsion. For a second academic offense the penalty is permanent expulsion."
A short list of some tautologies: 1. (p q) p 3. p (p q) 5. ~p (p q) 7. ((p q) p) q 9. ((p q) ~q) ~p
2. (p q) q 4. q (p q) 6. ~(p q) p 8. ((p q) ~p) q 10. ((p q) (q r)) (p r)
Mathematical induction: If P(n0) is true and assuming P(k) is true implies P(k+1) is true, then P(n) is true for all n > n0 Permutations and Combinations:
nPr
=
n! (n - r)!
nCr
=
n! r!(n - r)!
Properties of Relations: A relation is reflexive if a R a, for all a A. A relation is irreflexive if a R a, for all a A. A relation is symmetric if whenever a R b, then b R a. A relation is asymmetric if whenever a R b, then b R a. A relation is antisymmetric if whenever a R b and b R a, then a = b. A relation is transitive if whenever a R b and b R c, then a R c. A relation is called an equivalence relation if it is reflexive, symmetric, and transitive.
Each of the following six arguments uses one of the tautologies listed on the coversheet. (See table under the heading, "a short list of some tautologies.") For each of the four arguments, identify which tautology was used from this list by entering a value 1 through 10 in the space provided. (2 points each) 1. If it is thundering, then there is lightning There is thunder There is lightning Answer: ____7______ 3. Either Ed is short or Ed is tall Ed is not short Ed is tall Answer: ____8______ 5. This test is easy Either I studied well or this test is easy Answer: ____3 or 4__ 6. 4. 2. It is either raining or snowing It isn't raining It must be snowing Answer: ___8_______ If I drive to school, I will be late to class I am on time for class I didn't drive to school Answer: ____9______ Matthew is my son Matthew is a child of mine Answer: _1, 2, 3, or 4__
For the next four arguments, indicate which are valid and which are invalid. (2 points each) 7. If I publish a novel, I will be famous If I am famous, I will be happy I am happy, therefore, I published a novel Valid 9. Invalid 10. 8. If I drive to school, I will be late to class I was late to class I drove to school Valid Invalid
If I try hard, then I will succeed If I succeed, then I will be happy I am not happy, therefore, I didn't try hard Valid Invalid
Pete is the name of my pet The only dogs I own are black labs Pete is a black lab Valid Invalid
The following seven problems present seven situations where r items are selected from a set of n items. Select the formula, nr, nPr, nCr, or (n+r-1)Cr, that will compute the number of different, valid sequences and identify the values of r and n. (4 points each) To answer these problems, you simply need to remember which formula pertains to which situation: ordered/unordered and duplicates allowed/duplicates not allowed. Each situation is either ordered with duplicates allowed, nr, ordered with no duplicates allowed, nPr, unordered with no duplicates allowed, nCr, or unordered with duplicates allowed, (n+r-1)Cr. From there, its just a matter of setting n to the number of items in the set being selected from and r to the number of items being selected. 11. Compute the number of 4-digit ATM PINs where duplicate digits are allowed. a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____10___
r = ____4____
12. Compute the number of different 5 card hands can be drawn from a deck of 52 cards. a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____52___
r = ____5____
13. How many committees of 5 people can be created from a group of 8 people? a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____8____
r = ____5____
14. How many ways can the letters in the word MICHAEL be arranged? a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____7____
r = ____7____
15. Assume you need to buy 10 bottles of soda from a selection of {Coke, Pepsi, Dr. Pepper, and Sprite}. How many ways could you do this? a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____4____
r = ____10___
16. How many three-digit numbers are there in base-5? Assume leading zeros are included as digits. a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____5____
r = ____3____
17. How many different ways can 2 six-sided dice come up? There is no order, e.g., 3 & 4 are the same as 4 & 3. a.) nr b.)
nPr
c.)
nCr,
d.)
(n+r-1)Cr
n = ____6____
r = ____2____
18. True or false: In selecting r items from a set of n items where order doesn't matter and duplicates are allowed, r may be greater than n. (2 points) 19. Which of the following expressions describes how to calculate the number of available license plate combinations of the format ABC 123? (2 points) a.) 26C3 10C3 e.) (36 + 6 1)C6 b.) (26 + 10 1)C10 f.) 263103 c.) 26P3 10P3 g.) 26C3 103 d.) (26 + 10 1)P6 h.) None of the above
20. Assume we have a lottery where you first pick 5 from a group of 60 then pick one powerball option from a group of 45? What is the ratio of picking the wrong powerball to picking the right powerball? points) (2 a.) 45:1 b.) 44:1 c.) 60C544 : 60C545 d.) 45C1 : 1 f.) None of the above
21. Let A = {a, b} and B = {1, 2, 3}. List all of the elements in A B. (3 points) A B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} 22. If |A| = 5 and |B| = 10, then the cardinality of A B is: (2 points) a.) 510 b.) 105 c.) 510 d.) 5C10 d.) 5P10 f.) None of the above
The next 5 problems represent relations across the Cartesian product A A where A = {a, b, c, d}. The relations are represented either as subsets of A A, matrices, or digraphs. For each problem, determine whether the relation is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, and/or transitive. Check all that apply. (4 points each) For each of the following answers, either the matrix or the digraph is used to determine the relations characteristics. Reflexive says that there must be ones all along the main diagonal of the matrix and every vertex in the digraph has an edge that loops back to itself. Irreflexive says that there must be zeros all along the main diagonal of the matrix and no vertex in the digraph has an edge that loops back to itself. Symmetric says that the matrix is symmetric across the main diagonal and for every edge in the digraph, there is an equal edge going back the other way, i.e., every edge is bidirectional (cycles of length one are allowed). Asymmetric says that for every 1 in the matrix, there must be a 0 opposite the main diagonal from it and there can be no 1s on the diagonal while every edge in the digraph must be only one-way and there can be no loop backs, i.e., cycles of length one. Antisymmetric says that for every 1 in the matrix, there must be a 0 opposite the main diagonal from it, but 1s are allowed on the diagonal. Every edge in the digraph must be only one-way, but loop backs, i.e., cycles of length one, are allowed. Transitive says that for every pair of 1s in a matrix aij and ajk, there must be a 1 at aik. The digraph says that for every path of length two, there must also be a path of length one.
23. R = {(a,b), (a,c), (a,d), (b,c) , (b,d) , (c,d)} Below, I have created both the matrix defining R and the digraph. a MR = 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 c reflexive 24. R = A A Below, I have created both the matrix defining R and the digraph. a MR = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 c reflexive irreflexive symmetric asymmetric antisymmetric d transitive b irreflexive symmetric asymmetric antisymmetric d transitive b
25.
0 1 0 1
1 0 1 1
0 1 0 0
1 1 0 1
a
b
c reflexive 26. 1 1 0 0 1 1 0 0 0 0 1 1 irreflexive 0 0 1 1 symmetric asymmetric a
d antisymmetric b transitive
c reflexive 27. a irreflexive b MR = symmetric asymmetric 0 0 0 0 1 0 0 1 1 0 0 1
d antisymmetric 0 0 0 0 transitive
c reflexive
d irreflexive symmetric asymmetric antisymmetric transitive
The next three problems represent relations across the Cartesian product A A where A = {1, 2, 3, 4, 5}. 28. Write the set of ordered pairs represented by the relation matrix (4 points) R = {(1, 1), (1, 3), (2, 1), (2, 5), (3, 2), (4, 5), (5, 2), (5, 5)} 29. Convert the following digraph to a matrix. (3 points) 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1
1 5 3
2
0 1 0 0 0
0 1 0 0 1
1 0 0 0 0
0 0 0 1 0
1 0 1 0 0
4
30. Fill out the table below listing the in-degree and out degree of each element for the digraph of the previous problem. (4 points) In-Degree Out-Degree 1 1 2 2 2 2 3 1 1 4 1 1 5 2 1
31. Create the digraph of the relation R = A A for the set A = {a, b, c}. (3 points) a b
c 32. Select only one of the following statements to prove true using mathematical induction. (7 points) a.) 2 + 4 + 6 + + 2n = n(n + 1) b.) 1 + 21 + 22 + 23 + + 2n = 2n+1 1 c.) 1 + a1 + a2 + a3 + + an1 = an 1 a1
a.) First, test to see if the base case is...