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Jim Hefferon
proofs
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6fad727136ac94810489e7f261ed38a5b16752d9...127b9e9b4c24d9365dfb2cc0a71465e618ea390a
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127b9e9b4c24d9365dfb2cc0a71465e618ea390a
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Commits (1)
hefferon_final submission
· 127b9e9b
Jim Hefferon
authored
Aug 08, 2018
127b9e9b
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ibl.tex
ibl.tex
+16
14
hefferon_final.tex
jiblm/hefferon_final.tex
+62
87
iblcompactmaxlength.pdf
output/iblcompactmaxlength.pdf
+0
0
iblmaxlength.pdf
output/iblmaxlength.pdf
+0
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preface.tex
preface.tex
+12
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ibl.tex
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127b9e9b
...
...
@@ 53,7 +53,8 @@ or the positive integers $\Z^+=\set{1,2,\ldots}$.
and~
$
k
$
is the
\definend
{
quotient
}
.
(Alternative wordings are:
$
d
$
~
\definend
{
is a factor of
}
~
$
n
$
,
or
$
d
$
~
\definend
{
goes evenly into
}
~
$
n
$
,
$
d
$
~
\definend
{
goes evenly into
}
~
$
n
$
,
$
n
$
~
\definend
{
is divisible by
}
~
$
d
$
,
or
$
n
$
~
\definend
{
is a multiple of
}
~
$
d
$
.)
We denote the relationship as
$
d
\divides
n
$
if
$
d
$
~is a divisor of~
$
n
$
...
...
@@ 63,8 +64,8 @@ or the positive integers $\Z^+=\set{1,2,\ldots}$.
\begin{df}
A number is
\definend
{
even
}
if it is divisible by~
$
2
$
,
otherwise it is
\definend
{
odd
}
.
(We may instead say that
a number has
\definend
{
even parity
}
or
\definend
{
odd parity
}
.)
(We may instead say that
the
\definend
{
parity
}
is even
or
odd
.)
\end{df}
The notation
$
d
\divides
n
$
signifies a relationship between two integers.
...
...
@@ 2074,7 +2075,7 @@ An object~$x$ that belongs to a set~$A$ is an \definend{element}
or
\definend
{
member
}
of that set, denoted
$
x
\in
A
$
.
Denote that
$
x
$
is not an element with~
$
x
\notin
A
$
.
Two s
ets are equal if and only if they have the same elements.
S
ets are equal if and only if they have the same elements.
% \end{df}
%\noindent
(Read `
$
\in
$
' as ``is an element of'' rather than ``in'' to avoid
...
...
@@ 2194,7 +2195,7 @@ Prove.
\end{problem}
\begin{problem}
\label
{
ex:PropertiesOfSubset
}
Prove, for
any
sets
$
A
$
,
$
B
$
, and~
$
C
$
.
Prove, for sets
$
A
$
,
$
B
$
, and~
$
C
$
.
\begin{exes}
\begin{exercise}
\notetext
{
Mutual Inclusion
}
If
$
A
\subseteq
B
$
and
$
B
\subseteq
A
$
then
$
A
=
B
$
.
...
...
@@ 2620,7 +2621,7 @@ where $X$ is the universe, $\universalset=X$.
% \end{answer}
% \end{problem}
\begin{problem}
\pord
\begin{problem}
\pord
.
\begin{exes}
% \begin{exercise}
% $AB\subseteq A$
...
...
@@ 3554,16 +3555,17 @@ This is asymmetric because the
definition puts no such condition on output elements.
\begin{df}
A function is
\definend
{
onetoone
}
, or
\definend
{$
1
$

$
1
$}
,
A function
$
\map
{
f
}{
D
}{
C
}$
is
\definend
{
onetoone
}
, or
\definend
{$
1
$

$
1
$}
,
or an
\definend
{
injection
}
,
if for each value there is at most
one associated argument, that is, if
$
f
(
d
_
0
)=
f
(
d
_
1
)
$
implies that
$
d
_
0
=
d
_
1
$
for all elements
$
d
_
0
,d
_
1
$
of the domain.
A function is
\definend
{
onto
}
, or a
\definend
{
surjection
}
if for each value there is at least
one associated argument, that is, if for each element
$
c
$
of the codomain
there exists an element
$
d
$
of the domain such that
$
f
(
d
)=
c
$
.
A function that is both onetoone and onto, so that for every value there
one associated argument, that is,
if
$
f
(
d
_
0
)=
f
(
d
_
1
)
$
for
$
d
_
0
,d
_
1
\in
D
$
implies that
$
d
_
0
=
d
_
1
$
.
The function is
\definend
{
onto
}
, or a
\definend
{
surjection
}
,
if for each member of the codomain
$
c
\in
C
$
there is at least
one associated argument, at least one
$
d
\in
D
$
such that
$
f
(
d
)=
c
$
.
A function that is both onetoone and onto, so that for every member
of the codomain there
is exactly one associated argument, is a
\definend
{
correspondence
}
or
\definend
{
bijection
}
.
\end{df}
...
...
jiblm/hefferon_final.tex
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preface.tex
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127b9e9b
...
...
@@ 6,18 +6,19 @@ Its only prerequisite is high school mathematics.
\smallskip
\noindent\textsc
{
Approach.
}
This course is
inquirybased (sometimes called discovery m
ethod or
Moore
m
ethod).
This course is
InquiryBased (sometimes called the Discovery M
ethod or
Moore
M
ethod).
Students get a sequence of exercises
along with definitions and a few remarks.
They work through the material together, by
proving statements or providing examples.
This means that each person must grapple directly with the
mathematics
\Dash
the instructor only
lightly guides and the students pledge not to use outside sources
\Dash
talking out misunderstandings,
sometimes stumbling in the dark, and sometimes
having beautiful flashes of insight.
They attempt these outside of class,
and then in class propose solutions.
The instructor only
lightly guides and the students pledge not to use outside sources so
they together talk through misunderstandings,
sometimes stumble in the dark, and sometimes
have beautiful flashes of insight.
Most important is that each person must grapple directly with the
mathematics.
For these students, with this material,
this is the best way to develop mathematical maturity.
Besides, it is a great deal of fun.
...
...
@@ 47,7 +48,7 @@ intellectual habits that we established at the start.
\noindent\textsc
{
Exercises.
}
Where possible,
nearby exercises have about the same difficulty.
This gradually rises.
This
level
gradually rises.
Some exercises have multiple items; these come in two types.
If the items are labeled
\textsc
{
A
}
,
\textsc
{
B
}
, etc.,
...
...