CMSC-203 Discrete Math Assignments (spring 2000)

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

• Problem 1: Exercise 15 (Set 5.1 on page 242)
• Problem 2: Exercise 39 (Set 5.2 on page 256)
• Problem 3: Exercise 39 (Set 5.3 on page 267)
• Problem 4: Exercise 48 (Set 5.3 on page 267)
• Problem 5: Exercise 6 (Set 5.4 on page 271) Your answer to Problem 5 should be in the form of an essay of at least 1/2 page in length.

Homework 2: Logic

• Problem 1: Exercise 50 (Set 1.1 on page 16)
• Problem 2: Exercises 30 and 32 (Set 1.2 on page 28)
• Problem 3: Exercises 41 and 43 (Set 1.3 on page 41)
• Problem 4: Exercises 14 and 15 (Set 1.4 on page 55)
• Problem 5: Exercise 34 (Set 1.5 on page 74)

Homework 3: Quantified Logic and Proofs

• Problem 1: Exercises 24-27 (Set 2.1 on page 88)
  Also express each statement in first-order predicate logic.
• Problem 2: Exercises 31-34 (Set 2.2 on page 98)
  Also, for each statement, express the statement in first-order predicate logic; give its converse, inverse, and contrapositive; and specify which of these implications are true.
• Problem 3: Exercise 19 (Set 2.3 on page 110)
• Problem 4:
  (a) Exercise 31 (Set 3.1 on page 125)
  (b) Exercise 31 (Set 3.4 on page 147)
  (c) Exercise 27 (Set 3.5 on page 153)
• Problem 5: Exercises 14-15 (Set 3.6 on page 161)

Homework 4: 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 30,34,38 (Set 4.1 on page 193)
• Problem 2: Exercise 13 (Set 4.2 on page 205)
• Problem 3: Exercise 20 (Set 4.3 on page 211)
• Problem 4: Exercise 8 (Set 4.4 on page 220)
• Problem 5: Exercise 7 (Set 4.5 on page 230)

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 "xmaple &". You can also obtain a free copy of Maple for your PC from UCS in the ECS building (level 100).
  (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 coeficients 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)
• Problem 5: Exercise 46 (Set 10.5 on page 601)

Part III: Counting

Focus on how to count, including by fundamental principles (addition and product rules), urn model (four cases), inclusion/exclusion, 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.

Homework 9: Counting

Carefully review Sections 7.4 and 7.6. and 8.1, focusing on all explanations and examples involving counting.

• Problem 1: Exercises 5 and 7 (Set 6.1 on page 279)
• Problem 2: Exercise 7 (Set 6.2 on page 293)
• Problem 3: Exercises 20 and 30 (Set 6.2 on pages 294-5)
• Problem 4: Exercises 7 and 29 (Set 6.3 on pages 303-6)
• Problem 5: Exercises 11 and 12 (Set 6.4 on page 321)

Homework 10: Counting

• Problem 1: Exercises 14 and 16 (Set 6.5 on page 329)
• Problem 2: Exercise 16 (Set 6.6 on page 336)
• Problem 3: Exercises 16 and 17 (Set 6.7 on page 343)
• Problem 4: Exercise 24 (Set 7.4 on page 400)
• Problem 5: Exercise 49 (Set 8.1 on page 441)

Homework 11: Counting

• Problem 1: Exercise 33 (Set 6.3 on page 306)
• Problem 2: Exercises 38 and 42 (Set 8.1 on page 441)
• Problem 3: Exercise 45 (Set 8.1 on page 441)
  (a) Solve problem as directed by text. (b) Also solve problem using inclusion/exclusion.
• Problem 4: Exercises 13 and 30 (Set 7.6 on page 423)
• Problem 5: Exercise 27 (Set 7.6 on page 423)

Bonus Homework: Difference Equations