CMSC-203 Discrete Math Assignments (spring 2001)

It is the foremost responsibility of each student to solve many problems each and every day--many more than are required to be handed in. Learning discrete math takes place primarily through solving problems actively -- not through passively reading nor passively listening to lectures.

Part I: Proofs

Focus on how to write proofs, including by counterexample, by direct proof, by contradiction, by contraposition, by mathematical induction (weak and strong forms), and by epsilon/delta proofs.

Homework 1: Sets and logic

• Problem 1: Exercises 36 and 39 (Set 5.2 on page 258)
• Problem 2: Exercise 33 (Set 5.3 on page 267)
• Problem 3: Exercises 32 and 48 (Set 1.1 on page 16)
• Problem 4: Exercises 41 and 43 (Set 1.3 on page 41)
• Problem 5: Exercises 27 and 29 (Set 2.2 on page 98)

Homework 2: Logic and proofs

• Problem 1: Exercise 24 (Set 3.1 on page 125)
• Problem 2: Exercise 26 (Set 3.2 on page 131
• Problem 3: Exercise 42 (Set 3.3 on page 139)
• Problem 4: Exercise 35 (Set 3.4 on page 147)
• Problem 5: Exercises 28 and 29 (Set 3.5 on page 153)

Homework 3: Induction

Whenever doing any induction proof in CMSC-203, always follow the style demonstrated in class. In particular, always begin with an EXPLICIT definition of the inductive set, typically called S. Clearly show the basis and inductive steps. In the inductive step, clearly identify the inductive hypothesis and where it is used. Also, always state whether you are using the weak or strong form of induction.

• Problem 1: Exercises 14 (Set 3.6 on page 161)
  
• Problem 2: Exercises 10 (Set 3.7 on page 166)
  
• Problem 3: Exercise 16 (Set 4.2 on page 205)
• Problem 4: Exercise 20 (Set 4.3 on page 211)
• Problem 5: Exercises 6 (Set 4.4 on page 220)

Homework 4: Induction and Loop Invariants

• Problem 1: Exercise 2 (Set 4.4 on page 219)
• Problem 2: Exercise 14 (Set 4.4 on page 220)
• Problem 3: Exercise 7 (Set 4.5 on page 230)
• Problem 4: Exercise 9 (Set 4.5 on page 230)
• Problem 5: Exercise 11 (Set 4.5 on page 230)

There is no homework due March 7 because Exam I on proofs is March 7.

Part II: Calculation

Focus on how to calculate, including solving summations and recurrences and using Maple.

Homework 5: Functions

Read the note " On functions." Focus on calculation; go lightly on the counting topics from Sections 7.4 and 7.6, which will be revisited in the final third of the course.

• Problem 1: Exercise 24 (Set 7.1 on page 356)
• Problem 2: Exercises 9-10 (Set 7.3 on page 385)
• Problem 3: Exercise 18-19 (Set 7.4 on page 400)
• Problem 4: Exercise 21 (Set 7.5 on page 411)

• Problem 5: Explore Maple. You can run Maple from UMBC mainframes by typing "maple" or "xmaple &". You can also purchase a copy of Maple for your PC or Mac from the UMBC Bookstore.
  (a) Work through the on-line Maple tutorial.
  (b) Consider the summations and products in Exercises 19-25 (Set 4.1 on page 193). For each Exercise, replace the upper bound with the integral variable N, and for consistency call the index variable i. Using Maple, solve each exercise in three ways:
  (i) The upper bound is the variable N.
  (ii) The upper bound is the particular constant value given.
  (iii) Graph each of your solutions (to variation (i)) in three dimensions by viewing the solution as a function of the two independent variables a and N, where a is the lower index bound (the sum or product ranges from i=a to N).
  Hand in a transcript of your Maple session, together with your graphs.

Homework 6: Recursion

See the reading assignment now listed under HW5. Also, go lightly on the counting examples from Section 8.1, which will be revisited in the final third of the course.

• Problem 1: Exercises 16 and 50 (Set 8.1 on pages 438-441). Use induction in Exercise 16.
• Problem 2: Exercise 45 (Set 8.2 on page 453)

• Problem 3: Exercises 12,14,15 (Set 8.3 on page 465).
  (a) Solve each recurrence by hand.
  (b) Solve each recurrence with Maple. Hand in a transcript of your Maple session.
  (c) Graph the solution to each recurrence in Maple as a function of the index parameter k.

• Problem 4: Exercise 24 (Set 8.3 on page 466)
• Problem 5: Exercise 22 (Set 8.4 on page 475). Use induction.

Homework 7: Linear Difference Equations

Read the handout on solving linear difference equations (handwritten notes by Dr. Sherman), which augments the book's explanation of linear difference equations by treating the cases of inhomogeneous equations and equations with periodic solutions.

Instructions: In Problems 1-4 below, solve each of the recurrences given in the specified exercises by hand. Express your answers as closed-form expressions in terms of the index parameter. Begin by restating each recurrence in standard form (with any and all inhomogeneous terms appearing on the right hand side, each written as a polynomial times an exponential). Check your answers using the initial conditions. Whenever these exercises ask you to perform some other task, disregard the book's instructions and simply solve the recurrences.

Whenever a recurrence has a periodic solution, express your solution in terms of the elementary periodic functions SIN and COS, as explained in the handout by Dr. Sherman; never express any real-valued solution using any coefficients nor bases that are non-real complex numbers.

• Problem 1: Exercises 2,6,8 (Set 8.1 on page 438)
• Problem 2: Exercises 6,8,9 (Set 8.2 on page 451)
• Problem 3: Exercises 9,10 (Set 8.3 on page 465)
• Problem 4: Exercises 22,23 (Set 8.3 on page 466). Read and follow the above instructions regarding periodic solutions and complex numbers.

• Problem 5: Solve all of the recurrences given in Problems 1-4 using Maple. Also using Maple, graph the solution to each recurrence as a function of the index parameter. Hand in a transcript of your Maple session together with your graphs.

Homework 8: Relations

• Problem 1: Exercises 24,27 (Set 10.1 on pages 545-546)
• Problem 2: Exercise 16,24,36 (Set 10.2 on pages 554-555)
• Problem 3: Exercise 35 (Set 10.3 on page 572)
• Problem 4: Exercise 11 (Set 10.5 on page 600) and Exercise 47 (Set 10.5 on page 601)
• Problem 5: Solve the special problem on RSA and Diophantine equations. See model solution to HW4, Problem 5 on Extended Euclid's Algorithm.

There is no homework due April 18 because Exam II on calculation is April 18.

Part III: Counting

Focus on how to count, including by fundamental principles (addition and product rules), urn model (four cases), inclusion/exclusion (general case with n sets), partition and sum, setting up summation or recurrence, and seat-of-the-pants estimates. Also learn proof by counting argument (including pigeonhole principle and by diagonalization).

Carefully review Sections 7.4 and 7.6. and 8.1, focusing on all explanations and examples involving counting. Read Chapter 6 and the handouts on inclusion/exclusion, proofs by counting arguments, and probability. Note that the textbook's treatments of inclusion/exclusion and diagonalization are weak and incomplete.

Motivating examples for this part of the course include: counting steps in programs, counting number of data structures or Boolean circuits of various types, separating complexity classes, the halting porblem, and determining the probability of success of a probabilistic program.

Playing and analyzing the game of backgammon can be very helpful in learning discrete probability. There are free on-line backgammon programs, such as Hans Berliner's BKG.

Homework 9: Counting

• Problem 1: Exercises 19 and 24 (Set 7.4 on page 400)
• Problem 2: Exercises 48 and 49 (Set 8.1 on page 441)
• Problem 3: Exercises 3 and 4 (Set 6.1 on page 279)
• Problem 4: Exercises 15 and 16 (Set 6.2 on page 293)
• Analyze the Monte Hall Problem as described in class. Answer and simulation

Homework 10: Counting

• Problem 1: Exercises 2 and 8 (Set 6.3 on page 303)
• Problem 2: Exercise 33 (Set 6.3 on page 306)
• Problem 3: Exercises 13 and 24 (Set 6.4 on pages 312-312)
• Problem 4: Exercises 9 and 13 (Set 6.5 on page 329)
• Problem 5: Exercises 18 and 20 (Set 6.6 on page 336)

Homework 11: Counting

• Problem 1: Exercises 14 and 15 (Set 7.6 on page 423)
• Problem 2: Exercises 24 and 27 (Set 7.6 on page 423)
• Problem 3: Exercise 17 (Set 6.7 on page 343)
  
• Problem 4: Exercises 45 and 49 (Set 8.1 on page 441)
  You must solve these problems by inclusion/exclusion.
• Problem 5: Analyze the following Maryland lottery games: Lotto, Big Game, and Pick-3/4.
  Your answer should be in the form of an essay.

Bonus Homework: Difference Equations