Test and Exam Information
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Test 1
|
Test 2
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Test 3
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Final
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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.
Sections Covered:
1.1-1.8, 2.1-2.2, 2.3 (divisibility), 2.4 (Euclid's Algorithm),
3.2-3.3, 5.1, 6.1, 6.4
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, satisfiable,
unsatisfiable, translations between words and logic, functionally
complete sets of operators
- Predicate logic:
quantifiers, domain of discourse,
logical equivalences,
translations between words and logic
-
Sets:
notation (element, subset, empty-set, power set, etc),
cardinality (size of a set),
power set, set operations (union, intersection, complement, etc),
bitstring representation, Venn diagrams, membership tables,
relationship to propositional logic
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Relations:
matrix representation, properties (reflexive, symmetric, transitive,
equivalence)
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Functions:
notation, domain, range, one-to-one, onto, inverse,
big O, sum notation, number of functions from A to B,
sequences and notation
-
Algorithms:
definition/properties, expressing in psuedo-code,
analyzing efficiency (Big O, time complexity, worst-case, best-case,
average-case)
-
Divisibility:
definition and notation for "a divides b",
basic properties,
primes, composites, primality testing, fundamental theorem of arithmetic,
divisibility tests (for 2, 5, 10, 3, 9)
greatest common divisor, least common multiple, Euclid's algorithm
-
Induction/Recursion:
Proving things by induction, expressing things recursively,
reading and writing recurrence relations,
unraveling recurrance relations
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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
(Table 5 on page 17, for example).
Scale for Test 1
| Point Range | Letter Grade |
|---|
| 73-81 (87) | A |
| 62-72 | B |
| 51-61 | C |
| 42-50 | D |
Primary Sections Covered:
2.3-2.5,
3.2-3.4,
4.1,4.3,
5.1-5.3,
Recurrence Notes,
RSA Notes.
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.
Thus it is much more likely that information about
sets and functions, Euclid's algorithm, etc. will appear than information
about truth tables.
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. 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.
-
Algorithms and their complexity:
definition/properties, expressing in psuedo-code,
analyzing efficiency (Big O, time complexity, worst-case, best-case,
average-case)
-
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);
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 (pseudo-random numbers, 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);
representations of integers (different bases, 2's complement
representation and its reletionship to modular arithmetic).
See also the notes on RSA and Number Theory
[pdf,
ps]
-
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.
See also the notes on Solving Recurrences
[pdf,
ps]
-
Counting:
Bijection Principle; Sum Rule (and variations); Product Rule;
Division Principle; Permutations; Combinations. (Pigeon Hole
Principle will not be covered on test.)
Extra Practice Problems
I have included on the homework page
a list of review problems for those of you who like to prepare by working
on problems. It is a long list, but you can select from it those things you
would like to work on. You can also get a sample of problems from
past quizzes and tests in
postscript or
pdf format.
These were prepared last year, but the material covered on this
year's test 2 is quite similar, so 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.
Scale for Test 2
| Point Range | Letter Grade |
|---|
| 69-79 (82) | A |
| 58-68 | B |
| 47-57 | C |
| 38-46 | D |
Primary Sections Covered:
2.6,
4.1-4.5, 4.7,
6.1-6.5, 7.1-7.6
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 (transitive closure especially);
identifying and using matrix inverses.
-
Counting:
Sum Rule;
Product Rule (and bijection principle);
combining rules, combinations, permutations, binomial coefficients,
Pascal's triangle, definition of finite probability and applications
involving counting problems; expected value of random variable;
complement law.
-
Relations:
properties of relations (reflexive, irreflexive, symmetric,
transitive); equivalence relations;
representations (matrix, graph, set/list);
operations on relations (inverse, complement, union, intersection, etc)
and proving properties of them;
transitive closure and Warshall's algorithm.
-
Graphs:
"flavors" of graphs (directed/undirected, loops, multiple edges, etc);
modeling situations;
definitions (adjacent, incident, degree, bipartite, isomorphic, etc);
representations (adjacency matrix and adjacency list);
relationship between degrees of vertices and number of edges;
graph connectivity; graph invariants;
Euler paths/circuits;
Hamilton paths/circuits;
Dijkstra's algorithm;
All-Pairs Shortest Paths (APSP) and Single-Source Shortest Paths (SSSP)
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. It is a long list, but you can select from it those things you
would like to work on. You can also get a sample of problems from
past quizzes and tests in
postscript or
pdf format.
Scale for Test 3
| Point Range | Letter Grade |
|---|
| 75-88 (90) | A |
| 64-75 | B |
| 53-63 | C |
| 42-52 | D |
The final exam will be 1:30 -- 4:30 pm on Thursday, December 13.
The final exam will cover material from the entire course. The topics
and sections listed above can serve as a guide as you prepare for the final.
Only a few additional comments are in order:
-
The formulas for divide-and-conquer (the so-called ``Master Theorem'')
will be provided on your exam. More details available in
[ps format]
[pdf format]
-
Be especially mindful of connections between topics -- especially between
topics that ocurred at different times in the course. Watch for
big ideas, recurring themes, common techniques, etc.
-
In case you misplaced one of your exams, here they are:
If I think of something else I want you to know, I'll post it here, so
check back. Be sure to look over your old tests, sample problems,
review sheets, and review problem sets as you prepare for the final.
There will be a review session 11:15--1:15 pm in NH 295 on Wednesday, December
12.
Scale for Final Exam
| Point Range | Letter Grade |
|---|
| 162-194 (173) | A |
| 141-161 | B |
| 120-140 | C |
| 100-119 | D |