REVIEW QUESTIONS Chapter 1: Foundations: Sets, Logic, and Algorithms 1. Why can t a Venn diagram be used to prove a statement about sets? 2. Suppose S is a set with n elements. Explain why the power set of S has 2 n elements. 3. Explain de Morgan's laws. That is, how does one complement the union of sets? 4. Explain why if A B then B A. 5. Explain how to show that an argument is valid. 6. If a statement P is a tautology, how can one easily build from P a statement which is a contradiction? Why does this method work? 7. If A is true and B is false, list three compound statements using A and B whose truth value will be false. 8. Explain why A B = A B, then B must be the empty set. 9. Explain how to represent a set A as a bit string. 10. Is it ever true that A X B = B X A? Explain your answer. 11. Why is a set a subset of itself? Why is the empty set a subset of every set? 12. If the truth table were to be constructed for a statement S that is composed of simple statements p, q, r, and t, how many cases would need to be considered? Explain your answer. 13. How does one negate a quantified statement in the form xp(x)? In the form xp(x)? 14. How does one prove a statement that involves if and only if? 15. How does one construct a proof by contradiction? Chapter 2: Integers and Mathematical Induction 1. How does one use the Euclidean algorithm to find the greatest common divisor of two integers? 2. Explain how the greatest common divisor and the least common multiple of two integers are connected. 3. Does the well ordering principle apply to the entire set of integers? Why or why not? 4. Explain why 3 must divide n 3 -n for all integers n. 5. Explain why 3n+1 and 5n+2 are relatively prime for all integers n. 6. Explain how to represent an integer in two s complement. 7. Is there a convenient way to switch between binary and hexadecimal representations of the same integer? Explain your answer. 8. Explain how the first principle of mathematical induction works. 9. In proving that 64 divides 9 n -8n-1 for all integers n 0 by mathematical induction, a crucial step in the inductive step of the proof involves rewriting 9 n+1-8(n+1)-1 in what form? 10. Explain a simple way to determine whether a given integer is a prime.
11. What does the Fundamental Theorem of Arithmetic state? 12. Explain why if p is a prime and p divides ab then p divides a or p divides b. Give a counterexample to show this statement is not true in general if p is not a prime. 13. Why must there be an infinite number of primes of the form 6n+5? 14. In hexadecimal representation, why are A, B, C, used? 15. Under what conditions does the Diophantine equation ax + by = c have solutions? Chapter 3: Relations and Posets 1. Explain why the number of relations on a set with n elements is 2 (n2). 2. Explain how an equivalence relation on a set S gives rise to a partition of S. 3. Explain how a partition of a set S gives rise to an equivalence relation on S. 4. Explain a process to find the transitive closure of a relation on an n-element set. 5. If D is the digraph representation of a relation R, what property must D have if the relation R is reflexive? Explain. 6. If D is the digraph representation of a relation R, what property must D have if the relation R is symmetric? Explain. 7. If D is the digraph representation of a relation R, what property must D have if the relation R is antisymmetric? Explain. 8. Explain why the relation R = { (a,b) Z X Z 7 divides (a-b) } is an equivalence relation. 9. A relation R on a set A is called irreflexive if for all a A, (a, a) R. Give an example of a relation on Z that is irreflexive and transitive but not symmetric. 10. Explain why the number of reflexive relations on a set with n elements is 2 n2 n. 11. How do equivalence relations and partial order relations differ? 12. Explain why the digraph of a partial order relation cannot contain a closed path (other than a loop). 13. Explain how one obtains a linear ordering on a finite partially ordered set A. 14. What is a relational database? Explain. 15. What are SQL queries? Explain. Chapter 4: Matrices and Closures of Relations 1. How does one form the Boolean product of two Boolean matrices? 2. How does one transpose a matrix? 3. What is the matrix I n? 4. Suppose A is an nxn matrix. What does AxI n equal? What does A + I n equal? 5. What is the matrix 0 n? 6. Suppose A is an nxn matrix. What does Ax0 n equal? What does A + 0 n equal? 7. Is a diagonal matrix necessarily symmetric? Explain. 8. Is a symmetric matrix necessarily diagonal? Explain. 9. For Boolean Matrices A and B, is it ever possible for A B = A B? Explain. 10. Suppose M R is the matrix of a relation on a set A. From M R how can one determine if R is reflexive? From M R how can one determine if R is symmetric? 11. What is the definition of Warshalls' algorithm?
Chapter 5: Functions 1. What is the key idea that separates a function from a general relation? 2. What does it mean to say a function f: A -> B is i. one-to-one? ii. onto B? 3. Provide a counterexample to the statement that function composition is commutative. 4. Suppose that what is the largest subset of the real numbers that could serve as the domain of f, if f is to be real valued? 5. Suppose f is a function from A into B. Is the relation f -1 always a function? Explain. 6. Provide a counterexample to the statement if : X -> Y and A and B are nonempty subsets of X then f(a B) = f(a) (B). 7. Is it ever true that [x] + [x] = 2x? Is it always true? 8. What does it mean to say a set is countable? 9. Are there sets that are not countable? 10. Explain how to find the sum of the first n terms of an arithmetic progression: a, a+d, a+2d, 11. Explain how to find the sum of the first n terms of a geometric progression: a, ar, ar2, 12. What would be the product of the first n terms of a geometric progression: a, ar, ar 2, 13. i. How can you determine if a sequence is an arithmetic progression? ii. How can you determine if a sequence is a geometric progression? 14. Given a Cayley table how can one determine if the operation is commutative? 15. Would the integers with the operation multiplication form a monoid? Explain. Chapter 6: Congruences 1. What does it mean to say that a is congruent to b modulo m? 2. Explain why an integer n is divisible by 9 if and only if the sum of the digits of n is divisible by 9 3. Consider the system Z 6 with addition, is this system a monoid? Explain. 4. What common error are ISBN's designed to detect? 5. When checking out of a grocery store what could be a reason why the clerk repeatedly passes an item under the scanner? 6. If two digits are interchanged in a UPC is the error detected? Explain. 7. Does the congruence ax b (mod m) always have a solution x? Explain. 8. Explain the Chinese Remainder Theorem. 9. When using a hash function what problem may arise? 10. What is the purpose of linear probing and double hashing? 11. Is it always possible to play a round robin tournament with n teams in n rounds? Explain. 12. Explain why if you know (p n )for every prime p, then you know for every positive integer n. ( is the Euler phi function.) 13. Suppose the gcd(n, 180) =1. Explain why 18 divides n 30-1. 14. What difficult problem makes the RSA code seem safe?
Chapter 7: Counting Principles 1. When would one use the addition principle for counting with two tasks T 1 and T 2? 2. When would one use the multiplication principle for counting with two tasks T 1 and T 2? 3. What is the number of ways of performing T 1 OR T 2, if n(t I ) is the number of ways of performing T I? 4. What is the pigeonhole principle? 5. How many arrangements are there of a set with 35 elements? Explain. 6. How many arrangements are there of 8 elements from a 35 element set? 7. What is the difference between an r-permutation of n letters and an r-combination of n letters? 8. Explain why C(n,r) = C(n,n-r). 9. Explain why the number of ways to arrange 5 A's and 7 B's is 10. Explain why the number of ways to arrange 5 A s, 7 B s and 9 C s is. C(21,5) C(16,7) C(9,9) =. 11. What is the number of ways to select r objects from n with repetitions allowed? 12. Explain why 2n =. 13. What is the identity used to generate Pascal's triangle? 14. How does one find the next largest permutation? Chapter 8: Discrete Probability 1. How does the probability of an event make a comparison between the events which are more likely to occur and the events which are less likely to occur? 2. When are two events said to be mutually exclusive? 3. Find the size of the event that a head and an odd face are turned up in an experiment of tossing a die and throwing a die together. 4. You have a weighted coin. Give examples of some possible probability assignments to head and tail. 5. Discuss about the idea behind the defining criterion of probability assignment. 6. A card is drawn from a well-shuffled deck of 52 cards. Find the odds in favor of drawing a queen or a black card. 8. Discuss the practical meaning of conditional probability and start with this point to come into the concept of independent events. 9. Find the probability of not getting 2 of throwing a die, given that the face turned up is even. 10. Which problems are of Bayesian type? 11. Discuss about the events of some random experiment which should be considered as F 1, F 2,., F k and E. 12. What happens if E is independent to each of F 1, F 2,., F k? 13. The events F 1, F 2,., F k are exhaustive means that either of F 1, F 2,., F k happens.
14. If we throw a fair die, there are six possible outcomes. Still some problems concerning this experiment can be solved by Bernoulli trial technique. Discuss the type of these problems. 15. A fair coin is tossed 50 times. We are to find the probability of getting exactly 16 heads. Compare this problem with the above. 16. The probability that a computer remains trouble-free during the warrantee period is 0.8. What is the probability that 8 of our newly purchased 10 computers will give us no trouble during the warrantee period? Chapter 9: Recurrence Relations 1. What is a recurrence relation for a sequence a 0, a 1, a 2,..., a n,...? 2. What are the initial conditions for a recurrence relation? 3. What is the Fibonacci sequence? 4. What is the iteration method for solving a recurrence relation? 5. Verify that s n = 7n +1 is a solution for the recurrence relation a n = a n-1 + 7 where a 0 = 1. 6. Verify that sn = 2 n+1-1 is a solution for the recurrence relation a n = 3a n-1-2 an-2 where a 1 = 3 and a 0 = 1. 7. Suppose that a n = d a n-1 where n = 1 and a 0 = A. Verify that s n = Ad n is a solution. 8. What is a linear homogeneous recurrence relation of order k with constant coefficients? 9. What is the characteristic equation of the recurrence relation a n = c 1 a n-1 + c 2 a n-2? 10. For the recurrence relation a n = c 1 a n-1 + c 2 a n-2 what is the form of the solution if the characteristic equation has distinct roots r 1 and r 2? 11. For the recurrence relation a n = c 1 a n-1 + c 2 a n-2 what is the form of the solution if the characteristic equation has a root r 1 of multiplicity two? 12. What is a linear nonhomogeneous recurrence relation of order k with constant coefficients? 13. For the recurrence relation a n - da n-1 = b n u where d, b, u are constant and b and u are nonzero, what is the form of the solution if b d? 14. For the recurrence relation a n - da n-1 = b n u where d, b, u are constant and b and u are nonzero, what is the form of the solution if b = d? 15. For the recurrence relation a n = da n-1 + b n (un + v) where d, b, u, and v are constant and b and u are nonzero, what is the form of the solution if b = d? Chapter 10: Algorithms and Time Complexity 1. What does it mean to say that the function f(x) is big-o of g(x)? 2. If f(n) = a m n m + a m-1 n m-1 + a m-2 n m-2 +... + a 1 n + a 0 with a m 0, then f(n) is big-o of what function? 3. What does it mean to say that the function f(x) is omega of g(x)? 4. What does it mean to say that the function f(x) is theta of g(x)? 5. If f(n) = a m n m + a m-1 n m-1 + a m-2 n m-2 +... + a 1 n + a 0 with am 0, then f(n) is theta of what function?
6. If the function f(n) is O(g(n)) and (g(n)), then what can be said about f(n)? 7. What does it mean to say a real valued function f is nondecreasing? 8. What does it mean to say that a function f(x) is smooth? 9. In a sequential search of n elements the average number of comparisons is: 10. In a binary search of n elements the average number of comparisons is: 11. If selection sort is used to sort a list of n elements then the number of key comparisons is: 12. If insertion sort is used to sort a list of n elements then the number of key comparisons is: 13. If merge sort is used to sort a list of n elements then the number of comparisons in the worst case is: 14. In Strassen s algorithm to multiply two n x n matrices, the time complexity is: 15. What is the purpose of the Chained Matrix Multiplication algorithm? Chapter 11: Graph Theory 1. Explain why the sum of the degrees of the vertices of a graph is equal to twice the number of edges. 2. Explain why in a graph the number of vertices of odd degree is even. 3. What does it mean to say a graph is bipartite? 4. What are Ramsey numbers? 5. What is the difference between a walk and a path? 6. What is the difference between a cycle and a circuit? 7. What is a perfect matching in a graph? 8. What is an adjacency matrix of a graph G with n vertices? 9. What is an incidence matrix of a graph G with n vertices and m edges? 10. If a connected graph is Eulerian then the degree of each vertex is even. What is the basic idea in the proof of this statement? 11. Define the relation R on the sets of graphs by G R H if and only if G is isomorphic to H. Then what type of relation is R? 12. What is the fundamental idea in Dijkstra s shortest path algorithm? 13. What is a topological ordering for a directed graph G? 14. If a graph can be colored in 4 or fewer colors does it follow that the graph is planar? 15. Let G be a connected planar graph with v vertices, e edges, and f faces. What equation relates these variables? Chapter 12: Trees and Networks 1. In a tree T can there be more than one path from a vertex u to a vertex v? Explain. 2. In a tree T must there be at least one path from a vertex u to a vertex v? Explain. 3. What does it mean to say that the tree T 1 is isomorphic to T 2? 4. What does it mean to say that a graph is acyclic? 5. Is a graph with 8 vertices and 7 edges necessarily a tree? Explain. 6. What is a leaf in a rooted tree? 7. What is a descendant in a rooted tree?
8. What is a binary tree? 9. What is the left subtree of a vertex v in a binary tree? 10. How can one describe a preorder traversal? 11. What is a minimal spanning tree of a graph G? 12. Describe the idea behind Prim s algorithm. 13. What is a transport network? 14. What is a flow? 15. What is the max-flow min-cut theorem? Chapter 13: Boolean Algebra and Combinatorial Circuits 1. What is a two element Boolean algebra? 2. What is a Boolean expression over B in the variables x 1, x 2, x 3,..., x n? 3. Are the Boolean expressions (x 1 x 2 ) and x 1 + x 2 equal? 4. What do the idempotent laws state? 5. What do the absorption laws state? 6. How does one form the dual of a Boolean expression? 7. What is the duality principle? 8. Is it true that a Boolean expression is equal to its dual? If not provide a counterexample. 9. If one has the truth table for a Boolean expression how does one determine the disjunctive normal form? 10. If one has the truth table for a Boolean expression how does one determine the conjunctive normal form? 11. What is an atom in a Boolean algebra? 12. How many Boolean algebras are there with 28 elements? 13. What is the purpose of a Karnaugh map? 14. How many cells are required for a K-map for six variables? 15. Draw the circuit that corresponds to the Boolean expression (x + z)y. Chapter 14: Finite Automata and Languages 1. What is a language on? 2. What is the Kleene star of X? 3. What is a deterministic finite automaton? 4. Let M be the DFA given by Q = {q 0, q 1, q 2 }, = {0, 1}, q0 is the initial state, F = {q 2 }, and is given by What is the transition diagram for M? 5. Let M be the DFA (Q,, q 0,, F). What is the extended transition function *? 6. What is the purpose of the extended transition function *? 7. What is a regular language? 8. Is every language a regular language? Explain.
9. How can one determine whether the language A accepted by a DFA with n states is infinite? 10. With what operations does R, the set of all regular languages on, form a Boolean algebra? 11. What is the difference between a DFA and a NDFA? 12. How does a FSM differ from a DFA? 13. Let G = (V N,, P, S) be a context free grammar. What does it mean to say that L is the language generated by G? 14. Let G = (V N,, P, S) be a context free grammar. What is a derivation tree in G? What is a regular grammar?