Index Index
absolute value, Definition Activity
additive identity, Item
additive inverse, Item
aleph, Definition
algebra of sets, Subsection
antisymmetric relation, Activity
arcsine function, Activity
arithmetic sequence, Exercise
arrow diagram, Subsection
associative laws
for real numbers, Item
for sets, Item
average
of a finite set of numbers, Progress Check
axiom, Paragraph
basis step, Subsection Paragraph Paragraph
biconditional statement, Subsection Beginning Activity
forms of, Subsection
proof of, Subsection
bijection, Definition
Binet's formula, Exercise
Cantor's diagonal argument, Progress Check
Cantor's Theorem, Subsection
Cantor, Georg, Subsection
Cantor-Schröder-Bernstein Theorem, Theorem
cardinal number, Definition
cardinality, Definition Definition Definition Definition Definition
\(\aleph_0\), Definition
\(\boldsymbol{c}\), Definition
finite set, Definition Subsection
natural numbers, Definition
Cartesian plane, Subsection
Cartesian product, Definition Beginning Activity
cases, proof using, Beginning Activity Subsubsection
Cauchy sequence, Exercise
chain rule, Subsection
choose-an-element method, Exercise Subsection Progress Check Subsection
circular relation, Activity
closed interval, Activity
closed ray, Activity
closure properties, Paragraph Progress Check
codomain, Beginning Activity Definition
Cohen, Paul, Item
common divisor, Definition
commutative laws
for real numbers, Item
for sets, Item
commutative operation, Exercise
complement of a set, Definition
complex numbers, Paragraph
composite function, Definition
composite number, Definition Beginning Activity Beginning Activity
composition of functions, Definition
inner function, Paragraph
outer function, Paragraph
compound interest, Activity
compound statement, Beginning Activity
conditional, Item
conditional statement, Beginning Activity Definition Subsection Subsection
conclusion, Beginning Activity Definition Paragraph
contrapositive of, Definition
converse of, Definition
forms of, Subsection
hypothesis, Beginning Activity Definition Paragraph
logical equivalencies, Subsection
negation, Subsection
truth table, Paragraph
congruence, Subsection Subsection Subsection
Division Algorithm, Subsection
congruence class, Definition
congruent modulo \(n\), Definition
conjunction, Item
connective, Beginning Activity
consecutive integers, Exercise
consecutive natural numbers, Activity
constant, Definition
construction method, Paragraph
constructive proof, Subsection Paragraph Subsubsection
contained (in a set), Definition
continuous, Exercise
continuum, Paragraph
Continuum Hypothesis, Item
contradiction, Definition Beginning Activity Subsection
contrapositive, Definition Subsection Subsection
converse, Definition
coordinates, Definition
corollary, Paragraph
countable set, Definition
countably infinite set, Definition
countably infinite sets
subsets of, Paragraph
union of, Theorem
counterexample, Item Example Paragraph Subsection Progress Check
De Moivre's Theorem, Activity
de Moivre, Abraham, Activity
De Morgan's Laws
for indexed family of sets, Theorem
for sets, Item
for statements, Paragraph
decimal expression
for a real number, Subsection
normalized form, Definition
decomposing functions, Subsection
definition, Beginning Activity Paragraph
by recursion, Paragraph
denumerable set, Definition
dependent variable, Paragraphs
derivative, Beginning Activity
diagonal, Beginning Activity
difference of two sets, Definition
digraph, Paragraph
Diophantine equation, Definition
linear in one variable, Definition
linear in two variables, Definition Section
Diophantus of Alexandria, Paragraph
direct proof, Beginning Activity Subsection Subsection Subsection
directed edge, Paragraph
directed graph, Paragraph
directed edge, Paragraph
vertex, Paragraph
disjoint, Definition
pairwise, Definition Paragraph
disjoint sets, Definition
disjunction, Item
distributive laws
for indexed family of sets, Paragraph
for real numbers, Item
for sets, Item
divides, Definition Progress Check
divides property, Exercise
divisibility test, Paragraph
for 11, Exercise
for 3, Exercise
for 4, Exercise
for 5, Exercise
for 9, Paragraph
Division Algorithm, Subsection Paragraphs
congruence, Subsection
using cases, Subsection
divisor, Definition
Dodge Ball, Beginning Activity Progress Check
domain
of a function, Beginning Activity Definition
of a relation, Definition
domino theory, Subsection
element-chasing proof, Subsection
empty set, Subsection
properties, Item
equal functions, Definition
equal sets, Definition
equality relation, Subsection
equation numbers, Paragraph
equivalence class, Definition
properties of, Paragraph
equivalence relation, Definition
equivalent sets, Definition Subsection
Euclid's Elements, Subsection
Euclid's Lemma, Subsection
Euclidean Algorithm, Subsection
even integer, Definition Beginning Activity
exclusive or, Paragraphs
existence theorem, Paragraph Subsection Paragraph
existential quantifier, Definition
Extended Principle of Mathematical Induction, Definition
factor, Definition
factorial, Definition Paragraph
family of sets, Paragraph
indexed, Definition
intersection, Definition
union, Definition
Fermat's Last Theorem, Activity
Fermat, Pierre, Activity
Fibonacci numbers, Beginning Activity
Fibonacci Quarterly, Paragraph
finite set, Definition Beginning Activity
properties of, Subsection Definition
function, Beginning Activity Definition
as set of ordered pairs, Subsection
bijective, Definition
codomain, Beginning Activity Definition
composite, Definition
composition, Definition
domain, Beginning Activity Definition
equality, Definition
injective, Definition
inverse of, Definition Activity
invertible, Paragraph
of two variables, Item
one-to-one, Definition
onto, Definition
piecewise defined, Activity
projection, Exercise
range, Definition
surjective, Definition
Fundamental Theorem
of Arithmetic, Theorem
of Calculus, Activity
future value, Activity
geometric sequence, Paragraph
geometric series, Paragraph
greatest common divisor, Definition
Gödel, Kurt, Item
idempotent laws for sets, Item
identity function, Progress Check Exercise Exercise Exercise
identity relation, Subsection
image
of a set, Definition
of a union, Item
of an element, Definition
of an intersection, Item
implication, Item
inclusive or, Paragraphs
increasing, strictly, Exercise
independent variable, Paragraphs
indexed family of sets, Definition
De Morgan's Laws, Theorem
distributive laws, Paragraph
intersection, Definition
union, Definition
indexing set, Definition
inductive assumption, Paragraph
inductive hypothesis, Paragraph
inductive set, Definition
inductive step, Subsection Paragraph Paragraph
infinite set, Definition Beginning Activity
initial condition, Paragraph
injection, Definition
inner function, Paragraph
integers, Paragraph Paragraph Subsection
consecutive, Exercise
system, Subsection
integers modulo \(n\), Definition
addition, Definition
multiplication, Definition
Intermediate Value Theorem, Paragraph
intersection
of a family of sets, Definition
of an indexed family of sets, Definition
of two sets, Definition
interval, Activity
Cartesian product, Progress Check
closed, Activity
half-open, Activity
open, Activity
inverse image of a set, Definition
inverse of a function, Definition Activity
inverse of a relation, Definition
inverse sine function, Activity
invertible function, Paragraph
irrational numbers, Paragraph Exercise Definition Paragraph
Law of Trichotomy, Exercise
least upper bound, Activity
lemma, Paragraph
Leonardo of Pisa, Subsection
linear combination, Definition
linear congruence, Activity
linear Diophantine equations, Definition Section
logical operator, Beginning Activity
logically equivalent, Definition
Lucas numbers, Exercise
magic square, Exercise
mapping, Definition
mathematical induction
basis step, Subsection Paragraph Paragraph
Extended Principle, Definition Definition
inductive step, Subsection Paragraph Paragraph
Principle, Definition
matrix, Exercise
determinant, Exercise
transpose, Exercise
modular arithmetic, Definition
modus ponens, Exercise
multiple, Definition
multiplicative identity, Item
multiplicative inverse, Item
natural numbers, Paragraph Paragraph Subsection
consecutive, Activity
necessary condition, Item
negation, Item
of a conditional statement, Subsection Subsection
of a quantified statement, Subsection Example
neighborhood, Exercise
normalized form
of a decimal expression, Definition
number
rational, Definition
octagon, Beginning Activity
odd integer, Definition Beginning Activity
one-to-one correspondence, Definition
one-to-one function, Definition
only if, Item
onto function, Definition
open interval, Activity
open ray, Activity
open sentence, Definition
ordered pair, Definition Activity
ordered triple, Activity
ordinary annuity, Activity
outer function, Paragraph
pairwise disjoint, Definition Paragraph
partition, Definition
pentagon, Beginning Activity
perfect square, Definition Activity
piecewise defined function, Activity
Pigeonhole Principle, Subsection Activity
polygon, Beginning Activity
diagonal, Beginning Activity
regular, Beginning Activity
power set, Definition Activity Beginning Activity
cardinality, Activity
predicate, Definition
preimage
of a set, Definition
of a union, Item
of an element, Definition
of an intersection, Item
prime factorization, Beginning Activity
prime number, Definition Beginning Activity Beginning Activity
prime numbers
distribution of, Item
twin, Item
Principle of Mathematical Induction, Definition
proof, Paragraph
biconditional statement, Subsection
by contradiction, Beginning Activity Paragraph Subsection
constructive, Paragraph Subsubsection
contrapositive, Subsection Subsection
element-chasing, Subsection
mathematical proof, Paragraph
non-constructive, Subsection Paragraph
using cases, Beginning Activity Subsubsection
proper subset, Definition
propositional function, Definition
Pythagorean Theorem, Progress Check
quadratic formula, Exercise
quadrilateral, Beginning Activity
quantifier, Beginning Activity Subsection
existential, Definition
universal, Definition
quotient, Item
range, Definition
of a relation, Definition
rational number, Definition
ray
closed, Activity
open, Activity
real number system, Paragraph
recurrence relation, Paragraph
recursive definition, Paragraph
regular polygon, Beginning Activity
relation, Definition
antisymmetric, Activity
circular, Activity
divides, Progress Check
domain, Definition
equality, Subsection
equivalence, Definition
identity, Subsection
inverse of, Definition
range, Definition
relative complement, Definition
relatively prime integers, Definition
remainder, Item
ring, Exercise
roster method, Paragraph
Second Principle of Mathematical Induction, Definition
sequence, Beginning Activity Definition
arithmetic, Exercise
geometric, Paragraph
set
complement, Definition
difference, Definition
equality, Definition
intersection, Definition
power, Definition
proving equality, Subsection
relative complement, Definition
roster method, Paragraph
union, Definition
set builder notation, Paragraph
set equality, Definition
set of divisors function, Exercise
set theory, Subsection
solution set, Paragraph Beginning Activity
statement, Beginning Activity Subsection
biconditional, Subsection Beginning Activity
compound, Beginning Activity
conditional, Beginning Activity Definition Subsection Subsection
subset
proper, Definition
subset (of a set), Definition
sufficient condition, Item
surjection, Definition
syllogism, Exercise
tautology, Definition Beginning Activity
theorem, Paragraph
triangle, Beginning Activity
Triangle Inequality, Item
truth set, Definition Beginning Activity Beginning Activity
truth table, Paragraph
Twin Prime Conjecture, Item
type 0 integer, Exercise
type 1 integer, Exercise
type 2 integer, Exercise
uncountable set, Definition
undefined term, Paragraph
union
of a family of sets, Definition
of an indexed family of sets, Definition
of two sets, Definition
unique factorization, Paragraph
universal quantifier, Definition
universal set, Definition
properties, Item
upper bound, Activity
variable, Definition
dependent, Paragraphs
independent, Paragraphs
Venn diagram, Beginning Activity
vertex, Paragraph
Wallis cosine formulas, Exercise
Wallis sine formulas, Exercise
Well Ordering Principle, Paragraph
Wiles, Andrew, Exercise
writing guidelines, Subsection Subsection Subsection Subsection Subsection Subsection Subsection Subsection Appendix
zero divisor, Exercise