Test and Exam Information
|
Test 1
|
Test 2
|
Test 3
|
Final
|
Special Circumstances
If there is some reason (such as a documented disability) that makes it
necessary to provide special accomodations for an exam, please contact
me before each exam so that such arrangements can be made.
Primary Sections Covered:
2.3, 2.6 (and App. A)
4.1-4.8, 6.1-6.6
Test 3 will focus primarily on material covered since the
last test, but you should not have completely forgotten the material
from the first two tests.
Topics
The following list is not intended to be exhaustive, but can serve
as a good reminder of the types of things you should know for the test.
Note that there is a more extensive list of terms and results at the
end of each chapter. There are a few things on those lists that we
either did not cover or did not emphasize, but for the most part, those
are very useful lists.
-
Matrix operations:
Boolean and integer; algorithms and their complexity;
shapes of matrices that can be multiplied;
connection to graphs and relations;
identifying and using matrix inverses;
repeated squaring.
-
Counting and Probability:
Sum Rule;
Product Rule (and bijection principle);
combining rules; combinations, permutations, binomial coefficients,
Pascal's triangle; MISSISSIPPI problem, donut problem;
lexicographical order of permutations and combinations;
pigeonhole principle;
definition of finite probability and applications
involving counting problems;
sum and product rules for probability;
conditional probability, independent events;
random variables, expected value (note: expected value does not
appear in text, but we covered it in class).
-
Graphs:
"flavors" of graphs (directed/undirected, loops, multiple edges, etc);
modeling situations;
definitions (adjacent, incident, degree, bipartite, isomorphic, subgraph,
planer, etc);
representations (especially adjacency matrix)
relationship between degrees of vertices and number of edges;
graph connectivity; graph invariants;
Euler paths/circuits;
Hamilton paths/circuits;
-
Representations of Integers:
Representations using different bases, conversion between bases,
algorithm to convert to base b using mod operator (and why it works),
arithmetic operations performed in other bases,
2's complement encoding of negative numbers (and connection to modular
arithmetic).
I will be including on the homework page
a list of review problems for those of you who like to prepare by working
on problems. You can also get a sample of problems from
past quizzes and tests:
These are the same files I posted before test 2. These sample problems
are not intended to cover all topics that could occur on this test.
Primary Sections Covered:
2.4, 2.5,
1.6, 5.1-5.3,
Recurrence Notes,
Number Theory Notes,
3.3,
4.1-4.5, Expected Value
Test 2 will focus primarily on material covered since the
last test, but you should not have completely forgotten the material
from the first test. In particular, material from the first test
is most likely to appear in ways related to things we have done since
the first test. Recall that we did not finished our study of
induction and recursion until shortly after Test 1, so that material
occurs again here.
Some "old" topics have been so integral to the new material
that I have listed them again in the topics below.
Topics
The following list is not intended to be exhaustive, but can serve
as a good reminder of the types of things you should know for the test.
Note that there is a more extensive list of terms and results at the
end of each chapter (Chapter Review and Chapter Self-Test).
There are a few things on those lists that we
either did not cover or did not emphasize, but for the most part, those
are very useful lists.
-
Induction/Recursion:
Proving things by induction; expressing things recursively;
reading and writing recurrence relations;
unraveling recurrance relations; geometric sums;
solving recurrences of special forms (divide-and-conquer
and linear homogeneous); analyzing performance of recursive code.
-
Algorithms and their complexity:
Big O, analyzing complexity of recursive code.
-
Number Theory (Modular Arithmetic):
definition, notation and basic properites of the relation "a divides b";
definition, notation and basic properites of the operation "a mod n";
definition, notation and basic properites of the relation "a = b (mod n)";
primes and composites (primality testing, fundamental theorem of arithmetic,
divisibility tests for 2, 5, 10, 3, 9, 11);
greatest common divisor (Euclid's algorithm, gcd lemma);
modular arithmetic system (arithemtic operations, inverses, solving
linear equations, Chinese Remainder Theorem, etc);
RSA public key cryptography (including Fermat's Little Theorem and
Chinese Remainder Theorem);
other applications (ISBN, etc. -- you do
not need to know specific details of these, but should be able to
work with them if the details are provided);
-
Counting and Probability:
Bijection Principle; Addition Principle (and variations);
Multiplication Principle (and Division Principle);
permutations; combinations; probability; conditional probability;
random variables; expected value
Extra Practice Problems
I am making available a number of problems that I have used in the
past on quizzes or tests.
These were prepared 2-4 years ago, and the material covered this
year is not exactly the same as it was in those years,
but these problems are probably still useful.
The topics covered on these problems are not intended to represent an
exhaustive list of topics that could appear on the test.
In particular, you will notice a lack of probability problems.
Sections Covered:
1.1-1.3, 1.6, 2.1-2.2,
2.4-2.6, 2.8, 3.5, 5.1, 5.2
(including notes on solving recurrence relations)
Topics
The following list is not intended to be exhaustive, but can serve
as a good reminder of the types of things you should know for the test.
Note that there is a more extensive list of terms and results at the
end of each chapter. There are a few things on those lists that we
either did not cover or did not emphasize, but for the most part, those
are very useful lists.
- Propositional logic:
logical connectives (and, or, not, etc),
truth tables, logical equivalences, tautology,
translations between words and logic, functionally
complete sets of operators [why we only need to have and, or, not]
- Predicate logic:
quantifiers, domain of discourse,
logical equivalences [especially those involving negation as on pages 21-22],
translations between words and logic
-
Sets:
notation (element, subset, empty-set, power set, etc),
cardinality (size of a set, bijection principle),
power set, set operations (union, intersection, complement, etc),
bitstring representation, Venn diagrams, membership tables,
relationship to propositional logic
-
Relations:
matrix representation, properties (reflexive, symmetric, transitive,
equivalence, partial order, total order), equivalence relations
and equivalence classes
-
Functions:
notation, domain, range, one-to-one, onto, inverse,
using definition of big O, working with big O,
sum and product notation, number of functions from A to B,
sequences and notation
-
Induction/Recursion:
Proving things by induction, expressing things recursively,
reading and writing recurrence relations,
unraveling recurrance relations, solving linear homogeneous recurrences
-
Miscellaneous:
counting arguments (sizes of certain sets, number of booean functions, etc.);
relationship between propositional logic and sets;
most important rules for manipulating logic and sets