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\(\def\d{\displaystyle} \def\course{Math 228} \newcommand{\f}[1]{\mathfrak #1} \newcommand{\s}[1]{\mathscr #1} \def\N{\mathbb N} \def\B{\mathbf{B}} \def\circleA{(-.5,0) circle (1)} \def\Z{\mathbb Z} \def\circleAlabel{(-1.5,.6) node[above]{$A$}} \def\Q{\mathbb Q} \def\circleB{(.5,0) circle (1)} \def\R{\mathbb R} \def\circleBlabel{(1.5,.6) node[above]{$B$}} \def\C{\mathbb C} \def\circleC{(0,-1) circle (1)} \def\F{\mathbb F} \def\circleClabel{(.5,-2) node[right]{$C$}} \def\A{\mathbb A} \def\twosetbox{(-2,-1.5) rectangle (2,1.5)} \def\X{\mathbb X} \def\threesetbox{(-2,-2.5) rectangle (2,1.5)} \def\E{\mathbb E} \def\O{\mathbb O} \def\U{\mathcal U} \def\pow{\mathcal P} \def\inv{^{-1}} \def\nrml{\triangleleft} \def\st{:} \def\~{\widetilde} \def\rem{\mathcal R} \def\sigalg{$\sigma$-algebra } \def\Gal{\mbox{Gal}} \def\iff{\leftrightarrow} \def\Iff{\Leftrightarrow} \def\land{\wedge} \def\And{\bigwedge} \def\entry{\entry} \def\AAnd{\d\bigwedge\mkern-18mu\bigwedge} \def\Vee{\bigvee} \def\VVee{\d\Vee\mkern-18mu\Vee} \def\imp{\rightarrow} \def\Imp{\Rightarrow} \def\Fi{\Leftarrow} \def\var{\mbox{var}} \def\Th{\mbox{Th}} \def\entry{\entry} \def\sat{\mbox{Sat}} \def\con{\mbox{Con}} \def\iffmodels{\bmodels\models} \def\dbland{\bigwedge \!\!\bigwedge} \def\dom{\mbox{dom}} \def\rng{\mbox{range}} \def\isom{\cong} \DeclareMathOperator{\wgt}{wgt} \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} \newcommand{\va}[1]{\vtx{above}{#1}} \newcommand{\vb}[1]{\vtx{below}{#1}} \newcommand{\vr}[1]{\vtx{right}{#1}} \newcommand{\vl}[1]{\vtx{left}{#1}} \renewcommand{\v}{\vtx{above}{}} \def\circleA{(-.5,0) circle (1)} \def\circleAlabel{(-1.5,.6) node[above]{$A$}} \def\circleB{(.5,0) circle (1)} \def\circleBlabel{(1.5,.6) node[above]{$B$}} \def\circleC{(0,-1) circle (1)} \def\circleClabel{(.5,-2) node[right]{$C$}} \def\twosetbox{(-2,-1.4) rectangle (2,1.4)} \def\threesetbox{(-2.5,-2.4) rectangle (2.5,1.4)} \def\ansfilename{practice-answers} \def\shadowprops{{fill=black!50,shadow xshift=0.5ex,shadow yshift=0.5ex,path fading={circle with fuzzy edge 10 percent}}} \newcommand{\hexbox}[3]{ \def\x{-cos{30}*\r*#1+cos{30}*#2*\r*2} \def\y{-\r*#1-sin{30}*\r*#1} \draw (\x,\y) +(90:\r) -- +(30:\r) -- +(-30:\r) -- +(-90:\r) -- +(-150:\r) -- +(150:\r) -- cycle; \draw (\x,\y) node{#3}; } \renewcommand{\bar}{\overline} \newcommand{\card}[1]{\left| #1 \right|} \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)


\(\Delta^k\)-constant, Paragraph
additive principle, Section Subsection
adjacent, Paragraph Item
antecedent, Subsection
argument, Chapter
arithmetic sequence, Section
biconditional, Item
bijection, Paragraph
binomial coefficients, Subsection
binomial identity, Paragraph
bipartite, Paragraph Item
bit string, Subsection
Brooks' Theorem, Theorem
Canadians, Paragraphs
cardinality, Item Paragraph
Cartesian product, Paragraph Subsection
cases, Paragraph
characteristic equation, Paragraph
characteristic roots, Subsection Subsection
chromatic index, Paragraph
chromatic number, Item Paragraph
clique, Paragraph
closed formula, Section
codomain, Paragraph
coloring, Paragraph
combination, Paragraph
complement, Paragraph
complete graph, Paragraph Item
composition, Exercise
conclusion, Subsection Chapter
conditional, Item
congruence, Subsection
conjunction, Item
connected, Paragraph Item
connectives, Paragraph
and, Item
if and only if, Item
implies, Item
not, Item
or, Item
consequent, Subsection
contradiction, Paragraph
contrapositive, Item
proof by, Paragraph
converse, Item
convex, Paragraph
counterexample, Paragraph
cycle, Item
De Morgan's laws, Subsection
deduction rule, Paragraph
degree, Item
derangement, Paragraph
difference, of sets, Paragraph
Diophantine equation, Paragraph
direct proof, Paragraph
disjoint, Paragraph
disjunction, Item
divides, Paragraph
divisibility relation, Subsection
Division algorithm, Subsection
Doctor Who, Paragraph
dodecahedron, Subsection
domain, Paragraph
double negation, Subsection
edges, Paragraph
empty set, Item
Euler path, Item
Euler's formula, Section
existential quantifier, Subsection
faces, Paragraph
factorial, Paragraph
Fibonacci sequence, Example Exercise Item Paragraph Paragraph
finite differences, Section
Four Color Theorem, Theorem
function, Paragraph
gcd, Paragraph
geometric sequence, Section
girth, Paragraph
Goldbach conjecture, Paragraph
graph, Item
greatest commond divisor, Paragraph
Hall's Marriage Theorem, Theorem
Hamilton path, Paragraph
Hanoi, Investigate!
hypothesis, Subsection
icosahedron, Subsection
if and only if, Item
if…, then…, Item
implication, Item
inclusive or, Paragraph
induced subgraph, Section
induction, Paragraph Subsection
strong induction, Subsection
inductive hypothesis, Item
injection, Paragraph
integers, Item Item
intersection, Item Paragraph
inverse image, Paragraph
isomorphic, Paragraph Section
isomorphism, Section
isomorphism class, Paragraph
iteration, Paragraph
k-permutation, Section
Königsberg, Paragraph
lattice path, Paragraph
law of logic, Example
logical equivalence, Subsection
logically valid, seelaw of logic
magic chocolate bunnies, Exercise
matching, Paragraph
matching condition, Section
mod, Paragraph
modular arithmetic, Subsection
modulo \(n\), Subsection
modus ponens, Paragraph
monochromatic, Paragraph
multigraph, Paragraph Item
multiplicative principle, Section Subsection
natural numbers, Item
necessary condition, Subsection
negation, Item
neighbors, Paragraph
NP-complete, Paragraph
octahedron, Subsection
one-to-one, Paragraph
onto, Paragraph
partial sums, Paragraph
Pascal's triangle, [image]
perfect graph, Paragraph
permutation, Section
Petersen graph, Exercise
PIE, Paragraph
Pigeonhole principle, Example
planar, Item Paragraph
Platonic solids, Investigate!
polyhedron, Paragraph
polynomial fitting, Section
power set, Item Paragraph
premises, Chapter
prime numbers, Example Theorem
principle of inclusion/exclusion, Paragraph
product notation, Paragraph
proof by cases, Paragraph
proof by contradiction, Paragraph
proof by contrapositive, Paragraph
proposition, Paragraph
quantifiers, Subsection
exists, Subsection
for all, Subsection
Ramsey theory, Paragraph
range, Paragraph
rationals, Item
reals, Item
recurrence relation, Section
recursive definition, Section
reference, self, seeself reference
self reference, seereference, self
sequence, Paragraph
set, Paragraph
set difference, Paragraph
Sigma notation, Paragraph
stars and bars, Section
statement, Paragraph
subgraph, Section Item
induced, Section
subset, Paragraph
sufficient condition, Subsection
summation notation, Paragraph
surjection, Paragraph
tautology, Paragraph
telescoping, Paragraph
tetrahedron, Subsection
Tower of Hanoi, Paragraph
tree, Item
triangular numbers, Example Paragraph
truth table, Subsection
truth value, Paragraph
union, Item Paragraph
universal quantifier, Subsection
valid, Chapter
Venn diagram, Paragraph Paragraph
vertex coloring, Item Paragraph
vertices, Paragraph
Vizing's Theorem, Theorem
walk, Item
weight, of a string, Item
word, Example