MA 240 Discrete Mathematics, Fall
2013, Instructor: Jeffrey
Horn GUIDE TO THE FINAL EXAM
GENERAL
-
SCOPE:
The final exam for MA 240 is
comprehensive. It will focus on the topics covered in our eight
homeworks and six quizzes, so the best way to study is to look over the
assignments and quizzes including your submissions and the solutions.
-
TIME:
I try to design the exam so that it is easily completed within one hour,
leaving students with plenty of time (total time allocated for NMU final
exams is 1 hour and 50 minutes) to go over the test, double-checking,
etc.
-
OPEN BOOK: The
final, like all of our assessments, is open book, open notes, open use of
computers, but you must do your own work! (Don't be tempted to
use your laptop to communicate with others (other humans, that is!). Help from anyone but
the instructor is forbidden and must be punished (pretty darn harshly) if
detected.
-
COMPUTER USE: A laptop
is not absolutely required to complete this test (which for reasons of
reliability will be pencil-and-paper. But it is a good idea to have
a laptop with you. You will be allowed to use it to look things up.
But once again, you must do your own work, so don't be tempted to use your
laptop to communicate with others (other humans, that is!).
-
FORMAT: Mostly
multiple choice and fill-in-the-blank. Maybe a few short answer (one or
two sentences) questions. Very much like
our quiz questions!
TOPICS
(not necessarily complete, but I tried!)
- PROPOSITIONAL LOGIC
- Truth Tables (up to four inputs). Proof by truth table
(enumeration)
- Logical Equivalence (proof by truth table, proof by boolean algebra
axioms)
- Translating between natural language (English) statements and prop.
logic statements.
- Boolean Expressions:
- Operators: Negation, Implication, AND, OR, NAND, NOR, XOR,
etc.
- Compound Boolean Expressions
- SOME TERMS: Tautology, Contradiction, Satisfiable,
Unsatisfiable, Inverse, Converse, Contrapositive
- DeMorgan's Laws (both of them!)
- PREDICATE LOGIC
- The two quantifiers: Universal and Existential (or "upsidedown
A" and "backwards E")
- Domain of Discourse
- Translating between natural language (English) statements and
quantified logic statements.
- DeMorgan's Laws (both of them!) for the two quantifiers
- Nested quantifiers (e.g., Everybody loves somebody!)
- BASIC COUNTING
- Know when to multiply and when to add!
- Decision trees.
- Little Gauss' formula: Sum of integers 1 to n is n(n+1)/2
- COMBINATORICS AND PERMUTATIONS
- P(n) permutations, P(n,r) r-permutations, C(n,r)
r-combinations
- Number of shortest paths through a grid. Number of such
paths through a single junction in the grid.
- Growth rates.
- GRAPHS AND TREES
- Basic Graph Notation: graph G = (V,E), where V is a set of
vertices and E is a set of edges.
- Graph Terms: Undirected and directed graphs, connected graph,
simple graph (i.e., no self-loops, no multiple edges between a pair of
vertices), circuit, degree of a vertex, etc.
- Euler Paths and Circuits
- Hamilton Paths and Circuits
- Graph Encodings: Adjacency List, Adjacency Matrix
- DAGs (Directed Acyclic Graphs), as discussed last day of class.
- Tree Topics
- OTHER TOPICS
- Inductive proof
- The Binary Hypercube (graphing it, labeling it, the combinatorics of
it, etc.)