CS 214-01
STUDY GUIDE FOR TEST 2
SPRING 2014
GENERAL COMMENTS: Read the discussions in each section and work as many problems as you can. Working problems is the
best way to learn to differentiate between the various principles discussed in the chapters. Although the explanations seem
simple, it isn’t always obvious how to solve a particular problem. Review your homework, class notes, and as much as possible
of the textbook. Starred bullet points (*) are the “study-me-last” set – you can expect some of this to appear on the test but
they are not the primary areas of emphasis.
CHAPTER 2.5: Recurrence Relations
Important concepts: recurrence relations, closed form solutions, expand-guess-verify method of finding closed form solutions.
(On Test 1, you were responsible for terminology from this section, now you should be able to do simple problems).

Definition of recurrence relation (from 2.4, page 130)
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Definition of closed-form solution. Remember that with a recurrence relation S(n) is a function of S(n-1), so to
calculate one term you need to know all the previous terms. A closed form solution is an equation in n, so you can
plug in n and get S(n) directly, without having to compute preceding terms. See discussion on p. 147.

Example 42, practice 21
CHAPTER 3.1: Sets
Important Concepts: Set notation, definitions of standard sets (e.g., N = set of all nonnegative integers), relationships between
sets (equality, set inclusion, intersection, …), operations on sets, Venn diagrams

Be able to interpret set descriptions such as those in Example 2 & Practice 3.

Given a description of several sets, S1, S2 , S3, …, be able to show understanding of set relationships by listing the
elements in S1  S2 and other set relationships.

Practice 7, Example 5

Given sets S1, S2 , be able to enumerate the members of the power set (Si) and the Cartesian product (cross
product) of S1  S2.

Definition of binary and unary operations on a set.

Examples on page 193 and practice problem 12

Familiarize yourself with the Basic Set Identities on page 197. If you need any of 1 - 3 on the test they will be
provided, but not 4 & 5 as they should be obvious. *

What does it mean to say that a set of numbers is denumerable? Countable? *

Exercises 30, 33, 47.
CHAPTER 3.2: Counting
Important Concepts: multiplication principle, addition principle, decision trees; how to use for counting the number of objects
in a finite set.

Know how and when to apply the multiplication principle. Be able to distinguish between multiplication principle
with and without repetitions.

Example 26, example 27, practice 22.

Example 28: formula for the number of elements in set S1  S2

Addition principle: separately, and combined with the multiplication principle.

Exercise 11, 28 – pages 220-221; 37-39, page 222; 72 – 73, pages 223.
CHAPTER 3.3: Principle of Inclusion and Exclusion; Pigeonhole Principle
Important Concepts: Inclusion & exclusion, pigeonhole principle.

Understand the theory behind the two- and three- set formula for inclusion & exclusion.

Practice 26, Example 40, Example 42.

Definition of pigeonhole principle.

Example 44

Exercises 2 & 9, page 231; exercise 19, page232; exercise 24, page 233.
CHAPTER 3.4: Permutations and Combinations
Important Concepts: Understanding of permutations and combinations, and how they are different.

Permutation problems involve counting the number of arrangements of a given set of objects, with or without
repetition –without unless otherwise stated. Be able to recognize problems that involve permutations

Example 48, 49; practice 31.
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


Difference between combinations and permutations (arrangements). Be able to tell the difference between a
problem that requires combinations and one that requires permutations.
Example 54 & 55, practice 32 & 34.
Formulas for special cases of permutations and combinations *
Exercises 5, 7, 16, 25.
CHAPTER 3.5: Probability
Important Concepts: Finite probability as a counting problem, probability distributions, conditional probability.

Events, sample spaces, and calculation of probability as a function of set size. See page 252.

Practice 39 and 40, Example 65

Probability distributions when all events are not equally likely.

Practice 41

Conditional probability: definition, definition of independent events.*

Exercise 1-5, 17-20, page 261
CHAPTER 4.1: Relations

Definitions of binary relations: on a set, on multiple sets.

Example 4, example 5, practice 1, practice 2

Properties of relations: reflexive, symmetric, transitive, antisymmetric.

Be able to recognize the presence or absence of these properties when presented with a set

Example 6, 7, practice 4

Definition of the closure of a relation. *

Definition of relations that are partially ordered; be able to recognize relations that satisfy requirements for partial
ordering (or that don’t satisfy them).

Practice 8, example 10

Be able to identify and recognize characteristics of a Haase diagram. *

Equivalence relations: know definition, be able to recognize relations as being (or not being) an equivalence relation.

Set partitioning: definition, relation to equivalence relations.

Practice 14

Exercise 1, 6, 12, 37
CHAPTER 4.2: Functions

Definition of function; know terminology such as domain, codomain, range, image, preimage

Practice 23, Example 29

Definition of onto, one-to-one (injective) and bijective functions.

Inverse and identity functions.

Exercises 1, 6, 9

See Table 4.2 (page 342) for all definitions of interest)
THE TEST WILL STOP HERE